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A Survey of Arithmetic Applicationsof Capacity Theory

Robert Rumely

Banff International Research StationThe Geometry, Algebra and Analysis of Algebraic Numbers

October 8, 2015

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The logarithmic capacity γ(E)

For a compact set E ⊂ C, the logarithmic capacity γ∞(E) is ameasure of size arising in potential theory, with applications inanalysis, probability, approximation theory, and arithmetic.

In analysis, the main distinction is between sets of capacity 0and sets of positive capacity. Sets of capacity 0 are extremelysmall. Finite sets and compact countable sets have capacity 0.Sets of capacity 0 have Lebesgue measure 0, but not allcompact sets of Lebesgue measure 0 have capacity 0; themiddle thirds Cantor set has positive capacity.

Sets of capacity 0 tend to be “invisible” to holomorphic andharmonic functions. Many results which hold for isolated points,like the Riemann Extension Theorem, remain valid for sets ofcapacity 0.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The logarithmic capacity γ(E)

For a compact set E ⊂ C, the logarithmic capacity γ∞(E) is ameasure of size arising in potential theory, with applications inanalysis, probability, approximation theory, and arithmetic.

In analysis, the main distinction is between sets of capacity 0and sets of positive capacity. Sets of capacity 0 are extremelysmall. Finite sets and compact countable sets have capacity 0.Sets of capacity 0 have Lebesgue measure 0, but not allcompact sets of Lebesgue measure 0 have capacity 0; themiddle thirds Cantor set has positive capacity.

Sets of capacity 0 tend to be “invisible” to holomorphic andharmonic functions. Many results which hold for isolated points,like the Riemann Extension Theorem, remain valid for sets ofcapacity 0.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The logarithmic capacity γ(E)

For a compact set E ⊂ C, the logarithmic capacity γ∞(E) is ameasure of size arising in potential theory, with applications inanalysis, probability, approximation theory, and arithmetic.

In analysis, the main distinction is between sets of capacity 0and sets of positive capacity. Sets of capacity 0 are extremelysmall. Finite sets and compact countable sets have capacity 0.Sets of capacity 0 have Lebesgue measure 0, but not allcompact sets of Lebesgue measure 0 have capacity 0; themiddle thirds Cantor set has positive capacity.

Sets of capacity 0 tend to be “invisible” to holomorphic andharmonic functions. Many results which hold for isolated points,like the Riemann Extension Theorem, remain valid for sets ofcapacity 0.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The logarithmic capacity γ(E)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

In arithmetic, the main distinctions are between sets of capacity< 1, = 1, and > 1. There is often a sharp break in arithmeticphenomena at capacity 1.

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

In arithmetic, the main distinctions are between sets of capacity< 1, = 1, and > 1. There is often a sharp break in arithmeticphenomena at capacity 1.

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

In arithmetic, the main distinctions are between sets of capacity< 1, = 1, and > 1. There is often a sharp break in arithmeticphenomena at capacity 1.

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of capacities

Here are some examples of capacities:

When E = D(a, r) is a disc, γ∞(E) = r .

When E = [a,b] ⊂ R is a segment, γ∞(E) = (b − a)/4.

When E = {x2

a2 + y2

b2 ≤ 1} is an ellipse, γ∞(E) = (a + b)/2.

For Mandelbrot set, γ∞(M) = 1.

For the Julia set of a polynomial a0 + · · ·+ adzd with d ≥ 2,γ∞(J ) = |ad |−1/(d−1)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

G(z,∞; E) = 0 for z ∈ E ;G(z,∞; E) is continuous on C, except possibly on a set ofcapacity 0 contained in ∂E ;G(z,∞; E) is harmonic in C\E ;There is a constant V∞(E) such thatG(z,∞; E) = log(|z|) + V∞(E) + o(1) when z →∞.

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

G(z,∞; E) = 0 for z ∈ E ;G(z,∞; E) is continuous on C, except possibly on a set ofcapacity 0 contained in ∂E ;G(z,∞; E) is harmonic in C\E ;There is a constant V∞(E) such thatG(z,∞; E) = log(|z|) + V∞(E) + o(1) when z →∞.

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

G(z,∞; E) = 0 for z ∈ E ;G(z,∞; E) is continuous on C, except possibly on a set ofcapacity 0 contained in ∂E ;G(z,∞; E) is harmonic in C\E ;There is a constant V∞(E) such thatG(z,∞; E) = log(|z|) + V∞(E) + o(1) when z →∞.

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Green’s functions

The best way to determine γ∞(E) is to guess its Green’sfunction G(z,∞; E):

When γ∞(E) > 0, the Green’s function is the unique functionwhich satisfies

Thenγ∞(E) = e−V∞(E) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of Green’s functions

For a disc D(a, r), G(z,∞; E) = log+(|(z − a)/r |).

For a segment [a,b] ⊂ R,

G(z,∞; E) = log+(∣∣√(z−a)/z−b)+1√

(z−a)/z−b)−1

∣∣).This is gotten by conformally mapping C\[a,b] to C\D(0,1).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Examples of Green’s functions

For a disc D(a, r), G(z,∞; E) = log+(|(z − a)/r |).

For a segment [a,b] ⊂ R,

G(z,∞; E) = log+(∣∣√(z−a)/z−b)+1√

(z−a)/z−b)−1

∣∣).This is gotten by conformally mapping C\[a,b] to C\D(0,1).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Polýa-Carlson Theorem

The earliest arithmetic application of capacity theory was givenby F. Carlson and G. Pol’ya:

Theorem (Carlson 1921, extended by Pol’ya in 1922 and 1928)

Let f (z) =∑∞

n=k anz−n ∈ Z[z] be a Laurent series with integercoefficients, which converges in a neighborhood of∞, andwhich has an analytic continuation to the complement of a setE ⊂ C with γ(E) < 1. Then f (z) is the expansion of a rationalfunction p(z)/q(z) ∈ Q(z).

For example, the series f (z) = 2z−3 + 4z−5 + 6z−7 + · · ·converges near∞ and has an analytic continuation to thecomplement of the segment [−1,1]. It is the Laurent expansionof F (z) = 2z/(1− z2)2.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Polýa-Carlson Theorem

The earliest arithmetic application of capacity theory was givenby F. Carlson and G. Pol’ya:

Theorem (Carlson 1921, extended by Pol’ya in 1922 and 1928)

Let f (z) =∑∞

n=k anz−n ∈ Z[z] be a Laurent series with integercoefficients, which converges in a neighborhood of∞, andwhich has an analytic continuation to the complement of a setE ⊂ C with γ(E) < 1. Then f (z) is the expansion of a rationalfunction p(z)/q(z) ∈ Q(z).

For example, the series f (z) = 2z−3 + 4z−5 + 6z−7 + · · ·converges near∞ and has an analytic continuation to thecomplement of the segment [−1,1]. It is the Laurent expansionof F (z) = 2z/(1− z2)2.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

In 1968 Raphael Robinson gave a partial converse to thePol’ya-Carlson theorem:

Theorem (Robinson, 1968)Let E ⊂ C be a compact set, stable under complex conjugation,with γ(E) > 1. Then there are infinitely many Laurent seriesf (z) =

∑∞n=k anz−n ∈ Z[z] which converge near∞ and have an

analytic continuation to the complement of E, which are notrational.

There is also a theorem of Pommerenke, which I could notlocate while preparing these slides, which says that thePol’ya-Carlson theorem does hold when γ(E) = 1, undersuitable conditions on the Laurent series and the boundary ∂E .(?? continuous extension to ∂E , ∂E in Lip(1/2), E connected)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

In 1968 Raphael Robinson gave a partial converse to thePol’ya-Carlson theorem:

Theorem (Robinson, 1968)Let E ⊂ C be a compact set, stable under complex conjugation,with γ(E) > 1. Then there are infinitely many Laurent seriesf (z) =

∑∞n=k anz−n ∈ Z[z] which converge near∞ and have an

analytic continuation to the complement of E, which are notrational.

There is also a theorem of Pommerenke, which I could notlocate while preparing these slides, which says that thePol’ya-Carlson theorem does hold when γ(E) = 1, undersuitable conditions on the Laurent series and the boundary ∂E .(?? continuous extension to ∂E , ∂E in Lip(1/2), E connected)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Fekete’s Theorem

Another early arithmetic application of capacity was given byM. Fekete:

Theorem (Fekete, 1923)Let E ⊂ C be a compact set which is stable under complexconjugation. If γ∞(E) < 1, there are only finitely manyalgebraic integers whose conjugates all belong to E.

For example, if E = D(0, r) with 0 < r < 1, so γ∞(E) = r ,then 0 is the only algebraic integer in E .

If E = [−1/2,1/2], so γ∞(E) = 1/4, again 0 is the onlyalgebraic integer with all its conjugates in E .If E = [−1,1], so γ∞(E) = 1/2, then −1,0,1 are the onlyalgebraic integers with all their conjugates in E .

If E = [−2r ,2r ] where 0 < r < 1, so γ∞(E) = r , the onlyalgebraic integers with all conjugates in E are the finitely manyroots of Chebyshev polynomials with |2 cos(2πk/n)| ≤ 2r .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Fekete’s Theorem

Another early arithmetic application of capacity was given byM. Fekete:

Theorem (Fekete, 1923)Let E ⊂ C be a compact set which is stable under complexconjugation. If γ∞(E) < 1, there are only finitely manyalgebraic integers whose conjugates all belong to E.

For example, if E = D(0, r) with 0 < r < 1, so γ∞(E) = r ,then 0 is the only algebraic integer in E .

If E = [−1/2,1/2], so γ∞(E) = 1/4, again 0 is the onlyalgebraic integer with all its conjugates in E .If E = [−1,1], so γ∞(E) = 1/2, then −1,0,1 are the onlyalgebraic integers with all their conjugates in E .

If E = [−2r ,2r ] where 0 < r < 1, so γ∞(E) = r , the onlyalgebraic integers with all conjugates in E are the finitely manyroots of Chebyshev polynomials with |2 cos(2πk/n)| ≤ 2r .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Fekete’s Theorem

Another early arithmetic application of capacity was given byM. Fekete:

Theorem (Fekete, 1923)Let E ⊂ C be a compact set which is stable under complexconjugation. If γ∞(E) < 1, there are only finitely manyalgebraic integers whose conjugates all belong to E.

For example, if E = D(0, r) with 0 < r < 1, so γ∞(E) = r ,then 0 is the only algebraic integer in E .

If E = [−1/2,1/2], so γ∞(E) = 1/4, again 0 is the onlyalgebraic integer with all its conjugates in E .If E = [−1,1], so γ∞(E) = 1/2, then −1,0,1 are the onlyalgebraic integers with all their conjugates in E .

If E = [−2r ,2r ] where 0 < r < 1, so γ∞(E) = r , the onlyalgebraic integers with all conjugates in E are the finitely manyroots of Chebyshev polynomials with |2 cos(2πk/n)| ≤ 2r .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Fekete’s Theorem

Another early arithmetic application of capacity was given byM. Fekete:

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Fekete’s Theorem

Another early arithmetic application of capacity was given byM. Fekete:

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Fekete’s Theorem

Another early arithmetic application of capacity was given byM. Fekete:

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Fekete-Szegö Theorem

The converse to Fekete’s theorem was found by Fekete and G.Szegö more than 30 years later.

Theorem (Fekete, 1923; Fekete-Szegö, 1955)Let E ⊂ C be a compact set which is stable under complexconjugation. If γ(E) < 1, there is a neighborhood U ⊃ E whichcontains only finitely many complete conjugate sets of algebraicintegers. If γ(E) ≥ 1, then every neighborhood U ⊃ E containsinfinitely many conjugate sets of algebraic integers.

Note that this combines a finiteness theorem and an existencetheorem.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Fekete-Szegö Theorem

The converse to Fekete’s theorem was found by Fekete and G.Szegö more than 30 years later.

Theorem (Fekete, 1923; Fekete-Szegö, 1955)Let E ⊂ C be a compact set which is stable under complexconjugation. If γ(E) < 1, there is a neighborhood U ⊃ E whichcontains only finitely many complete conjugate sets of algebraicintegers. If γ(E) ≥ 1, then every neighborhood U ⊃ E containsinfinitely many conjugate sets of algebraic integers.

Note that this combines a finiteness theorem and an existencetheorem.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Fekete-Szegö Theorem

Strengthening the hypotheses on E , one can require that theconjugate sets of algebraic integers belong to E itself:

Theorem (Fekete-Szegö, 1955)Let E ⊂ C be a compact set which is stable under complexconjugation, has a piecewise smooth boundary, and is theclosure of its interior. If γ(E) > 1, there are infinitely manyconjugate sets of algebraic integers in E.

For example, for the disc D(0, r), when r ≥ 1 then all the rootsof unity belong to D(0,1) ⊂ D(0, r).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Fekete-Szegö Theorem

Strengthening the hypotheses on E , one can require that theconjugate sets of algebraic integers belong to E itself:

Theorem (Fekete-Szegö, 1955)Let E ⊂ C be a compact set which is stable under complexconjugation, has a piecewise smooth boundary, and is theclosure of its interior. If γ(E) > 1, there are infinitely manyconjugate sets of algebraic integers in E.

For example, for the disc D(0, r), when r ≥ 1 then all the rootsof unity belong to D(0,1) ⊂ D(0, r).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Robinson’s Theorem on totally real Algebraic Integers

What about sets E ⊂ R?In a significant advance, Raphael Raphael proved

Theorem (Robinson, 1964)

Let E ⊂ R be a finite union of closed intervals. If γ(E) > 1,E contains infinitely many conjugate sets of algebraic integers.

For example, when E = [−2r ,2r ] with r ≥ 1, the roots of all theChebyshev polynomials belong to [−2,2] ⊂ E .However, when γ(E) = 1 it is rare (even for intervals[a− 2,a + 2] with arbitrary a ∈ R) and in general it is a hardopen problem to determine when E contains infinitely manyconjugate sets of algebraic integers.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Robinson’s Theorem on totally real Algebraic Integers

What about sets E ⊂ R?In a significant advance, Raphael Raphael proved

Theorem (Robinson, 1964)

Let E ⊂ R be a finite union of closed intervals. If γ(E) > 1,E contains infinitely many conjugate sets of algebraic integers.

For example, when E = [−2r ,2r ] with r ≥ 1, the roots of all theChebyshev polynomials belong to [−2,2] ⊂ E .However, when γ(E) = 1 it is rare (even for intervals[a− 2,a + 2] with arbitrary a ∈ R) and in general it is a hardopen problem to determine when E contains infinitely manyconjugate sets of algebraic integers.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Robinson’s Theorem on totally real Algebraic Integers

What about sets E ⊂ R?In a significant advance, Raphael Raphael proved

Theorem (Robinson, 1964)

Let E ⊂ R be a finite union of closed intervals. If γ(E) > 1,E contains infinitely many conjugate sets of algebraic integers.

For example, when E = [−2r ,2r ] with r ≥ 1, the roots of all theChebyshev polynomials belong to [−2,2] ⊂ E .However, when γ(E) = 1 it is rare (even for intervals[a− 2,a + 2] with arbitrary a ∈ R) and in general it is a hardopen problem to determine when E contains infinitely manyconjugate sets of algebraic integers.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Uniform Approximation by Polynomials with IntegerCoefficients

A third application of capacities involves approximation ofcontinuous functions on an interval by polynomials with integercoefficients.

Before stating the theorem, let us consider some examples. LetE = [−1/2,1/2] and let f : E → R be continuous. Weierstrass’stheorem says f can be uniformly approximated by polynomialsin R[x ]. Suppose however, that we had uniform approximationby polynomials P1(z),P2(x), . . . ∈ Z[x ]. Since each polynomialtakes an integer value at x = 0, and since the valuesP1(0),P2(0), · · · converge uniformly to f (0), it must be thatf (0) ∈ Z and that Pn(0) = f (0) for all large n.

Theorem (Integer Polynomial Approximation on [-1/2,1/2)

. ] If f : [−1/2,1/2]→ R is continuous, then f can be uniformlyapproximated by polynomials in Z[x ] if and only if f (0) ∈ Z.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Uniform Approximation by Polynomials with IntegerCoefficients

A third application of capacities involves approximation ofcontinuous functions on an interval by polynomials with integercoefficients.

Before stating the theorem, let us consider some examples. LetE = [−1/2,1/2] and let f : E → R be continuous. Weierstrass’stheorem says f can be uniformly approximated by polynomialsin R[x ]. Suppose however, that we had uniform approximationby polynomials P1(z),P2(x), . . . ∈ Z[x ]. Since each polynomialtakes an integer value at x = 0, and since the valuesP1(0),P2(0), · · · converge uniformly to f (0), it must be thatf (0) ∈ Z and that Pn(0) = f (0) for all large n.

Theorem (Integer Polynomial Approximation on [-1/2,1/2)

. ] If f : [−1/2,1/2]→ R is continuous, then f can be uniformlyapproximated by polynomials in Z[x ] if and only if f (0) ∈ Z.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Uniform Approximation by Polynomials with IntegerCoefficients

A third application of capacities involves approximation ofcontinuous functions on an interval by polynomials with integercoefficients.

Before stating the theorem, let us consider some examples. LetE = [−1/2,1/2] and let f : E → R be continuous. Weierstrass’stheorem says f can be uniformly approximated by polynomialsin R[x ]. Suppose however, that we had uniform approximationby polynomials P1(z),P2(x), . . . ∈ Z[x ]. Since each polynomialtakes an integer value at x = 0, and since the valuesP1(0),P2(0), · · · converge uniformly to f (0), it must be thatf (0) ∈ Z and that Pn(0) = f (0) for all large n.

Theorem (Integer Polynomial Approximation on [-1/2,1/2)

. ] If f : [−1/2,1/2]→ R is continuous, then f can be uniformlyapproximated by polynomials in Z[x ] if and only if f (0) ∈ Z.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Uniform Approximation by Polynomials with IntegerCoefficients

Next suppose E = [−1,1]. By similar considerations, one seesthat necessarily f (−1), f (0), f (1) ∈ Z. Furthermore, for anyP(z) ∈ Z[x ], one has P(−1) ≡ P(1) (mod 2). Hence it must bethat f (−1) ≡ f (1) (mod 2) as well.

Theorem (Integer Polynomial Approximation on [-1,1)

. ] Let f : [−1,1]→ R be continuous. Then f can be uniformlyapproximated by polynomials in Z[x ] if and only iff (−1), f (0), f (1) ∈ Z and f (−1) ≡ f (1) (mod 2).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Uniform Approximation by Polynomials with IntegerCoefficients

Next suppose E = [−1,1]. By similar considerations, one seesthat necessarily f (−1), f (0), f (1) ∈ Z. Furthermore, for anyP(z) ∈ Z[x ], one has P(−1) ≡ P(1) (mod 2). Hence it must bethat f (−1) ≡ f (1) (mod 2) as well.

Theorem (Integer Polynomial Approximation on [-1,1)

. ] Let f : [−1,1]→ R be continuous. Then f can be uniformlyapproximated by polynomials in Z[x ] if and only iff (−1), f (0), f (1) ∈ Z and f (−1) ≡ f (1) (mod 2).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Uniform Approximation by Polynomials with IntegerCoefficients

The general theorem is as follows:

Theorem (?L. Ferguson)

Let E = [a,b] ⊂ R, and let C([a,b]) be the space of continuous,real-valued function on E. Then:(A) If γ(E) < 1, let A = {α1, α2, . . . , αn} ⊂ E be the finite set ofalgebraic integers whose conjugates all belong to E, given byFekete’s theorem. Let f ∈ C([a,b]). Then f can be uniformlyapproximated by polynomials in Z[x ] if and only if its values atthe points in A are matchable by those of a polynomial in Z[x ]:there is a P(x) ∈ Z[x ] such that f (α) = P(α) for each α ∈ A.(B) If γ(E) > 1, then Z[x ] is discrete in C([a,b]) for the L∞

norm. Hence there are functions f ∈ C([a,b]) which cannot beuniformly approximated by polynomials in Z[x ].

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Bilu’s Equidistribution Theorem

A more recent application of capacity theory is Bilu’s theorem.Let h : Q→ R≥0 be the absolute Weil height, which has thelocal decomposition

h(α) =∑

all places p of Q

1[Q(α) : Q]

∑σ:Q(α)↪→Cp

log+(|σ(α)|)p) .

Let µ = dθ/(2π) be the uniform measure of mass 1 on the unitcircle C(0,1), viewed as a singular measure on C.

Theorem (Bilu, 1997)

Let α1, α2, α3, · · · ∈ Q be a sequence of numbers for which[Q(αn) : Q]→∞ and h(αn)→ 0. For each n, let δn be thediscrete probability measure supported equally on theconjugates of αn. Then the measures δn converge weakly to µas n→∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Bilu’s Equidistribution Theorem

A more recent application of capacity theory is Bilu’s theorem.Let h : Q→ R≥0 be the absolute Weil height, which has thelocal decomposition

h(α) =∑

all places p of Q

1[Q(α) : Q]

∑σ:Q(α)↪→Cp

log+(|σ(α)|)p) .

Let µ = dθ/(2π) be the uniform measure of mass 1 on the unitcircle C(0,1), viewed as a singular measure on C.

Theorem (Bilu, 1997)

Let α1, α2, α3, · · · ∈ Q be a sequence of numbers for which[Q(αn) : Q]→∞ and h(αn)→ 0. For each n, let δn be thediscrete probability measure supported equally on theconjugates of αn. Then the measures δn converge weakly to µas n→∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Bilu’s Equidistribution Theorem

Although Bilu’s original proof used Fourier analysis, a smallobservation suggests that the theorem has a capacity-theoreticnature: log+(|z|) is the Green’s function of the unit discD(0,1) ⊂ C.

Let E ⊂ C be an arbitrary compact set, stable under complexconjugation, with capacity 1. Let hE : Q→ R≥0 be the “heightattuned to E” gotten by replacing each archimedean termlog+(|σ(α)|) in the decomposition of h(α) with G(σ(α),∞; E):

hE (α) =1

[Q(α) : Q]

∑σ:Q(α)↪→C

G(σ(α),∞; E)

+∑

finite places p of Q

1[Q(α) : Q]

∑σ:Q(α)↪→Cp

log+(|σ(α)|)p .

The distributional Laplacian of G(z,∞; E) is a probabilitymeasure supported on ∂E ⊂ C.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Bilu’s Equidistribution Theorem

Although Bilu’s original proof used Fourier analysis, a smallobservation suggests that the theorem has a capacity-theoreticnature: log+(|z|) is the Green’s function of the unit discD(0,1) ⊂ C.

Let E ⊂ C be an arbitrary compact set, stable under complexconjugation, with capacity 1. Let hE : Q→ R≥0 be the “heightattuned to E” gotten by replacing each archimedean termlog+(|σ(α)|) in the decomposition of h(α) with G(σ(α),∞; E):

hE (α) =1

[Q(α) : Q]

∑σ:Q(α)↪→C

G(σ(α),∞; E)

+∑

finite places p of Q

1[Q(α) : Q]

∑σ:Q(α)↪→Cp

log+(|σ(α)|)p .

The distributional Laplacian of G(z,∞; E) is a probabilitymeasure supported on ∂E ⊂ C.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Bilu’s Equidistribution Theorem

Theorem (R., 1999)Let E ⊂ C be a compact set of capacity 1, stable undercomplex conjugation. Let α1, α2, α3, · · · ∈ Q be a sequence ofnumbers for which [Q(αn) : Q]→∞ and hE (αn)→ 0. For eachn, let δn be the discrete probability measure supported equallyon the conjugates of αn. Then the measures δn convergeweakly to µ as n→∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Towards a Modern Theory of Capacities

We will now skip over a considerable amount of history, andpresent the modern, multi-center adelic theory of capacities.For each of the theorems above, there is an adelic versionwhich holds on any smooth projective curve over a global field.

It is good to regard a set E ⊂ C as belonging to P1(C). For anarbitrary ζ ∈ P1(C)\E , the Green’s function G(z, ζ; E) can bedefined, with properties as before but relative to ζ.

Fixing a uniformizer z − ζ, one defines the Robin constantrelative to ζ by

Vζ(E) = limz→ζ(G(z, ζ; E) + log(|z − ζ|)),and the capacity relative to ζ by

γζ(E) = e−Vζ(E).

More generally, for an arbitrary smooth complete curve C/C,one can define Green’s functions and capacities on C(C)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Towards a Modern Theory of Capacities

We will now skip over a considerable amount of history, andpresent the modern, multi-center adelic theory of capacities.For each of the theorems above, there is an adelic versionwhich holds on any smooth projective curve over a global field.

It is good to regard a set E ⊂ C as belonging to P1(C). For anarbitrary ζ ∈ P1(C)\E , the Green’s function G(z, ζ; E) can bedefined, with properties as before but relative to ζ.

Fixing a uniformizer z − ζ, one defines the Robin constantrelative to ζ by

Vζ(E) = limz→ζ(G(z, ζ; E) + log(|z − ζ|)),and the capacity relative to ζ by

γζ(E) = e−Vζ(E).

More generally, for an arbitrary smooth complete curve C/C,one can define Green’s functions and capacities on C(C)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Towards a Modern Theory of Capacities

We will now skip over a considerable amount of history, andpresent the modern, multi-center adelic theory of capacities.For each of the theorems above, there is an adelic versionwhich holds on any smooth projective curve over a global field.

It is good to regard a set E ⊂ C as belonging to P1(C). For anarbitrary ζ ∈ P1(C)\E , the Green’s function G(z, ζ; E) can bedefined, with properties as before but relative to ζ.

Fixing a uniformizer z − ζ, one defines the Robin constantrelative to ζ by

Vζ(E) = limz→ζ(G(z, ζ; E) + log(|z − ζ|)),and the capacity relative to ζ by

γζ(E) = e−Vζ(E).

More generally, for an arbitrary smooth complete curve C/C,one can define Green’s functions and capacities on C(C)

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Towards a Modern Theory of Capacities

Fixing a uniformizer z − ζ, one defines the Robin constantrelative to ζ by

Vζ(E) = limz→ζ(G(z, ζ; E) + log(|z − ζ|)),and the capacity relative to ζ by

γζ(E) = e−Vζ(E).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities and Green’s functions of p-adic sets

One also has capacities and Green’s functions for sets inP1(Cp),

For a disc Dp(a, r) = {z ∈ Cp : |z − a|b ≤ r} ⊂ Cp,G(z,∞; E) = log+(|z − a|p/r),

Vp,∞(E) = − logp(r),γp,∞(E) = r .

For the set of p-adic integers Zp, if µp is Haar measure on Zp,G(z,∞; Zp) =

∫Zp

logp(|z − w |p)dµp(w) + 1p−1 ,

Vp,∞(Zp) = 1p−1 ,

γp,∞(Zp) = p−1/(p−1).

With some work, for any smooth complete curve C/Cp, one candefine Green’s functions and capacities on C(Cp).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities and Green’s functions of p-adic sets

One also has capacities and Green’s functions for sets inP1(Cp),

For a disc Dp(a, r) = {z ∈ Cp : |z − a|b ≤ r} ⊂ Cp,G(z,∞; E) = log+(|z − a|p/r),

Vp,∞(E) = − logp(r),γp,∞(E) = r .

For the set of p-adic integers Zp, if µp is Haar measure on Zp,G(z,∞; Zp) =

∫Zp

logp(|z − w |p)dµp(w) + 1p−1 ,

Vp,∞(Zp) = 1p−1 ,

γp,∞(Zp) = p−1/(p−1).

With some work, for any smooth complete curve C/Cp, one candefine Green’s functions and capacities on C(Cp).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities and Green’s functions of p-adic sets

One also has capacities and Green’s functions for sets inP1(Cp),

For a disc Dp(a, r) = {z ∈ Cp : |z − a|b ≤ r} ⊂ Cp,G(z,∞; E) = log+(|z − a|p/r),

Vp,∞(E) = − logp(r),γp,∞(E) = r .

For the set of p-adic integers Zp, if µp is Haar measure on Zp,G(z,∞; Zp) =

∫Zp

logp(|z − w |p)dµp(w) + 1p−1 ,

Vp,∞(Zp) = 1p−1 ,

γp,∞(Zp) = p−1/(p−1).

With some work, for any smooth complete curve C/Cp, one candefine Green’s functions and capacities on C(Cp).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities and Green’s functions of p-adic sets

One also has capacities and Green’s functions for sets inP1(Cp),

For a disc Dp(a, r) = {z ∈ Cp : |z − a|b ≤ r} ⊂ Cp,G(z,∞; E) = log+(|z − a|p/r),

Vp,∞(E) = − logp(r),γp,∞(E) = r .

For the set of p-adic integers Zp, if µp is Haar measure on Zp,G(z,∞; Zp) =

∫Zp

logp(|z − w |p)dµp(w) + 1p−1 ,

Vp,∞(Zp) = 1p−1 ,

γp,∞(Zp) = p−1/(p−1).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Single-center Adelic Capacity

For each place p of Q, let Ep ⊂ Cp be a bounded closed set,stable under the group of continuous automorphismsAutc(Cp/Qp) ∼= Gal(Qp/Qp). For all but finitely many p, assumethat Ep = Dp(0,1) = {z ∈ Cp : |z|p ≤ 1}. Put

E =∏

all places p of Qγp,∞(Ep)

Define the global capacity γ(E,∞) =∏

p γp,∞(Ep).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example

For example, fix a finite set of primes S, let E∞ = [a,b] at thearchimedean place, and let Ep = Zp for each p. Then

γ(E,∞) =b − a

4·∏p∈S

p−1/(p−1) .

Theorem (An Adelic Fekete/Fekete-Szegö Theorem)With E as above,(A) If γ(E,∞) < 1 there are only finitely many points of Q(necessarily algebraic integers) whose archimedeanconjugates belong to [a,b], whose p-adic conjugates belong toZp for each p ∈ S, and whose p-adic conjugates for p /∈ Sbelong to Dp(0,1).(B) If γ(E,∞) > 1 there are infinitely many.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example

For example, fix a finite set of primes S, let E∞ = [a,b] at thearchimedean place, and let Ep = Zp for each p. Then

γ(E,∞) =b − a

4·∏p∈S

p−1/(p−1) .

Theorem (An Adelic Fekete/Fekete-Szegö Theorem)With E as above,(A) If γ(E,∞) < 1 there are only finitely many points of Q(necessarily algebraic integers) whose archimedeanconjugates belong to [a,b], whose p-adic conjugates belong toZp for each p ∈ S, and whose p-adic conjugates for p /∈ Sbelong to Dp(0,1).(B) If γ(E,∞) > 1 there are infinitely many.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities relative to multiple centers

What about the conjugates in Cp for p /∈ S?They all belong to Dp(0,1) = {z ∈ Cp : |z|p ≤ 1}.

An algebraic number is an algebraic integer if and only if itsp-adic conjugates belong to Dp(0,1) for all p.

Another way of viewing the integrality condition is to say thatthe conjugates avoid∞ in P1(Cp) for all finite primes. Note thatDp(0,1) = P1(Cp)\B(∞,1)−.

By allowing more general sets at nonarchimedean places, onecan construct algebraic numbers which satisfy prescribedconditions at finitely many places, and are integral at theremaining places.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities relative to multiple centers

What about the conjugates in Cp for p /∈ S?They all belong to Dp(0,1) = {z ∈ Cp : |z|p ≤ 1}.

An algebraic number is an algebraic integer if and only if itsp-adic conjugates belong to Dp(0,1) for all p.

Another way of viewing the integrality condition is to say thatthe conjugates avoid∞ in P1(Cp) for all finite primes. Note thatDp(0,1) = P1(Cp)\B(∞,1)−.

By allowing more general sets at nonarchimedean places, onecan construct algebraic numbers which satisfy prescribedconditions at finitely many places, and are integral at theremaining places.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities relative to multiple centers

What about the conjugates in Cp for p /∈ S?They all belong to Dp(0,1) = {z ∈ Cp : |z|p ≤ 1}.

An algebraic number is an algebraic integer if and only if itsp-adic conjugates belong to Dp(0,1) for all p.

Another way of viewing the integrality condition is to say thatthe conjugates avoid∞ in P1(Cp) for all finite primes. Note thatDp(0,1) = P1(Cp)\B(∞,1)−.

By allowing more general sets at nonarchimedean places, onecan construct algebraic numbers which satisfy prescribedconditions at finitely many places, and are integral at theremaining places.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities relative to multiple centers

What about the conjugates in Cp for p /∈ S?They all belong to Dp(0,1) = {z ∈ Cp : |z|p ≤ 1}.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Capacities relative to multiple centers

What about the conjugates in Cp for p /∈ S?They all belong to Dp(0,1) = {z ∈ Cp : |z|p ≤ 1}.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example

Up to now we have discussed numbers whose conjugates“avoid∞”; numbers which “avoid” other points are alsointeresting.

An algebraic number is a unit if it belongs toUp = {z ∈ Cp : |z|p = 1}, for each finite place p. Alternately, it“avoids” both 0 and∞ at all finite places.

Theorem (Robinson, 1968)

Let 0 < a < b ∈ R. Then the interval [a,b] contains infinitelymany totally real algebraic units if and only if

1 log(b−a4 ) > 0 and

2 log(b−a4 ) · log(b−a

4ab )− log(√

b+√

a√b−√

a)2 > 0

If either condition fails, there are only finitely many.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example

Up to now we have discussed numbers whose conjugates“avoid∞”; numbers which “avoid” other points are alsointeresting.

An algebraic number is a unit if it belongs toUp = {z ∈ Cp : |z|p = 1}, for each finite place p. Alternately, it“avoids” both 0 and∞ at all finite places.

Theorem (Robinson, 1968)

Let 0 < a < b ∈ R. Then the interval [a,b] contains infinitelymany totally real algebraic units if and only if

1 log(b−a4 ) > 0 and

2 log(b−a4 ) · log(b−a

4ab )− log(√

b+√

a√b−√

a)2 > 0

If either condition fails, there are only finitely many.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example

Up to now we have discussed numbers whose conjugates“avoid∞”; numbers which “avoid” other points are alsointeresting.

An algebraic number is a unit if it belongs toUp = {z ∈ Cp : |z|p = 1}, for each finite place p. Alternately, it“avoids” both 0 and∞ at all finite places.

Theorem (Robinson, 1968)

Let 0 < a < b ∈ R. Then the interval [a,b] contains infinitelymany totally real algebraic units if and only if

1 log(b−a4 ) > 0 and

2 log(b−a4 ) · log(b−a

4ab )− log(√

b+√

a√b−√

a)2 > 0

If either condition fails, there are only finitely many.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Discussion

An algebraic number is a unit if and only if its conjugatesbelong to

Dp(0,1)\Dp(0,1)− = P1(Cp)\(B(∞,1)− ∪ B(0,1)−)

for each p, that is, if it avoids∞ and 0 at each finite place.The conditions in the Theorem are equivalent to the negativedefiniteness of

Γ =

− log(b−a4 ) log(

√b+√

a√b−√

a)

log(√

b+√

a√b−√

a) − log(b−a

4ab ) .

There are Green’s functions and Robin constants with respectany point not in E . Here

Γ = Γ(E , {∞,0}) =

(V∞(E) G(0,∞; E)

G(∞,0; E) V0(E)

)is the ‘Green’s matrix of E ’ with respect to∞ and 0.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Discussion

An algebraic number is a unit if and only if its conjugatesbelong to

Dp(0,1)\Dp(0,1)− = P1(Cp)\(B(∞,1)− ∪ B(0,1)−)

for each p, that is, if it avoids∞ and 0 at each finite place.The conditions in the Theorem are equivalent to the negativedefiniteness of

Γ =

− log(b−a4 ) log(

√b+√

a√b−√

a)

log(√

b+√

a√b−√

a) − log(b−a

4ab ) .

There are Green’s functions and Robin constants with respectany point not in E . Here

Γ = Γ(E , {∞,0}) =

(V∞(E) G(0,∞; E)

G(∞,0; E) V0(E)

)is the ‘Green’s matrix of E ’ with respect to∞ and 0.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Discussion

An algebraic number is a unit if and only if its conjugatesbelong to

Dp(0,1)\Dp(0,1)− = P1(Cp)\(B(∞,1)− ∪ B(0,1)−)

for each p, that is, if it avoids∞ and 0 at each finite place.The conditions in the Theorem are equivalent to the negativedefiniteness of

Γ =

− log(b−a4 ) log(

√b+√

a√b−√

a)

log(√

b+√

a√b−√

a) − log(b−a

4ab ) .

There are Green’s functions and Robin constants with respectany point not in E . Here

Γ = Γ(E , {∞,0}) =

(V∞(E) G(0,∞; E)

G(∞,0; E) V0(E)

)is the ‘Green’s matrix of E ’ with respect to∞ and 0.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The General Framework

Let K be a global field, a number field or a finite extension ofFp(t) for some p. Fix an algebraic closure K of K .

Let C/K be a smooth, projective, geometrically integral curve.

Let X = {x1, . . . , xm} ⊂ C(K ) be a finite, galois-stable set ofpoints: the points to avoid.

For each place v of K , let Ev ⊂ C(Cv ) be a nonempty setdisjoint from X. We will require that Ev be galois-stable, andthat it be a finite union of ‘v -basic sets’ as defined below.

For all but finitely many places, we require thatEv = C(Cv )\(

⋃mi=1 B(xi ,1)−) be ‘X-trivial’.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The General Framework

Let K be a global field, a number field or a finite extension ofFp(t) for some p. Fix an algebraic closure K of K .

Let C/K be a smooth, projective, geometrically integral curve.

Let X = {x1, . . . , xm} ⊂ C(K ) be a finite, galois-stable set ofpoints: the points to avoid.

For each place v of K , let Ev ⊂ C(Cv ) be a nonempty setdisjoint from X. We will require that Ev be galois-stable, andthat it be a finite union of ‘v -basic sets’ as defined below.

For all but finitely many places, we require thatEv = C(Cv )\(

⋃mi=1 B(xi ,1)−) be ‘X-trivial’.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The General Framework

Let K be a global field, a number field or a finite extension ofFp(t) for some p. Fix an algebraic closure K of K .

Let C/K be a smooth, projective, geometrically integral curve.

Let X = {x1, . . . , xm} ⊂ C(K ) be a finite, galois-stable set ofpoints: the points to avoid.

For each place v of K , let Ev ⊂ C(Cv ) be a nonempty setdisjoint from X. We will require that Ev be galois-stable, andthat it be a finite union of ‘v -basic sets’ as defined below.

For all but finitely many places, we require thatEv = C(Cv )\(

⋃mi=1 B(xi ,1)−) be ‘X-trivial’.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The General Framework

Let C/K be a smooth, projective, geometrically integral curve.

Let X = {x1, . . . , xm} ⊂ C(K ) be a finite, galois-stable set ofpoints: the points to avoid.

For all but finitely many places, we require thatEv = C(Cv )\(

⋃mi=1 B(xi ,1)−) be ‘X-trivial’.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The General Framework

Let C/K be a smooth, projective, geometrically integral curve.

Let X = {x1, . . . , xm} ⊂ C(K ) be a finite, galois-stable set ofpoints: the points to avoid.

For all but finitely many places, we require thatEv = C(Cv )\(

⋃mi=1 B(xi ,1)−) be ‘X-trivial’.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Cantor Capacity

For each place v , define the local Green’s matrix to be them ×m symmetric matrix

Γ(Ev ,X) =

Vx1(Ev ) G(x2, x1; Ev ) · · · G(xm, x1; Ev )

G(x1, x2; Ev ) Vx2(Ev ) · · · G(xm, x2; Ev )...

.... . .

...G(x1, xm; Ev ) G(x2, xm; Ev ) · · · Vxm (Ev )

If X ⊂ C(K ), put E =

∏v Ev . Define the global Green’s matrix

Γ(E,X) =∑

v

Γ(Ev ,X) log(Nv) ,

where Nv is the order of the residue field at v , and log(Nv) = 1if Kv ∼= R and log(Nv) = 2 if Kv ∼= C.

If X 6⊂ C(K ), put L = K (X) and let Γ(E,X) = 1[L:K ] Γ(EL,X).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Cantor Capacity

For each place v , define the local Green’s matrix to be them ×m symmetric matrix

Γ(Ev ,X) =

Vx1(Ev ) G(x2, x1; Ev ) · · · G(xm, x1; Ev )

G(x1, x2; Ev ) Vx2(Ev ) · · · G(xm, x2; Ev )...

.... . .

...G(x1, xm; Ev ) G(x2, xm; Ev ) · · · Vxm (Ev )

If X ⊂ C(K ), put E =

∏v Ev . Define the global Green’s matrix

Γ(E,X) =∑

v

Γ(Ev ,X) log(Nv) ,

where Nv is the order of the residue field at v , and log(Nv) = 1if Kv ∼= R and log(Nv) = 2 if Kv ∼= C.

If X 6⊂ C(K ), put L = K (X) and let Γ(E,X) = 1[L:K ] Γ(EL,X).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Cantor Capacity

For each place v , define the local Green’s matrix to be them ×m symmetric matrix

Γ(Ev ,X) =

Vx1(Ev ) G(x2, x1; Ev ) · · · G(xm, x1; Ev )

G(x1, x2; Ev ) Vx2(Ev ) · · · G(xm, x2; Ev )...

.... . .

...G(x1, xm; Ev ) G(x2, xm; Ev ) · · · Vxm (Ev )

If X ⊂ C(K ), put E =

∏v Ev . Define the global Green’s matrix

Γ(E,X) =∑

v

Γ(Ev ,X) log(Nv) ,

where Nv is the order of the residue field at v , and log(Nv) = 1if Kv ∼= R and log(Nv) = 2 if Kv ∼= C.

If X 6⊂ C(K ), put L = K (X) and let Γ(E,X) = 1[L:K ] Γ(EL,X).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Cantor Capacity

LetPm = {(s1, . . . , sm) ∈ Rm : s1, . . . , sm ≥ 0, s1 + · · ·+ sm = 1}denote the set of m-element probability vectors.

There is a simple criterion for a symmetric m ×m matrix tobe negative definite: The value of Γ as a matrix game is

val(Γ) = max~s∈Pm

min~r∈Pm

t~sΓ~r ,

and Γ is negative definite if and only if val(Γ) < 0.

In general, for E =∏

v Ev and X = {x1, . . . , xm}, the Cantorcapacity of E with respect to X is defined to be

γ(E,X) = e− val(Γ(E,X)) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Cantor Capacity

LetPm = {(s1, . . . , sm) ∈ Rm : s1, . . . , sm ≥ 0, s1 + · · ·+ sm = 1}denote the set of m-element probability vectors.

There is a simple criterion for a symmetric m ×m matrix tobe negative definite: The value of Γ as a matrix game is

val(Γ) = max~s∈Pm

min~r∈Pm

t~sΓ~r ,

and Γ is negative definite if and only if val(Γ) < 0.

In general, for E =∏

v Ev and X = {x1, . . . , xm}, the Cantorcapacity of E with respect to X is defined to be

γ(E,X) = e− val(Γ(E,X)) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Cantor Capacity

LetPm = {(s1, . . . , sm) ∈ Rm : s1, . . . , sm ≥ 0, s1 + · · ·+ sm = 1}denote the set of m-element probability vectors.

There is a simple criterion for a symmetric m ×m matrix tobe negative definite: The value of Γ as a matrix game is

val(Γ) = max~s∈Pm

min~r∈Pm

t~sΓ~r ,

and Γ is negative definite if and only if val(Γ) < 0.

In general, for E =∏

v Ev and X = {x1, . . . , xm}, the Cantorcapacity of E with respect to X is defined to be

γ(E,X) = e− val(Γ(E,X)) .

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

The Fekete-Szegö Theorem with Local RationalityConditions

Theorem (R, 2012)

Let K be a global field. Let C/K be a smooth, projective,geometrically integral curve. Let X = {x1, . . . , xm} ⊂ C(K ) be afinite set of points stable under Aut(L/K ). For each place v ofK , let Ev ⊂ C(Cv )\X be a nonempty set which is a finite unionof v-basic sets and is stable under the group of continuousautomorphisms Autc(Cv/Kv ) ∼= Aut(K sep/Kv ). Assume that Evis X-trivial for all but finitely many v.

Put E =∏

v Ev . If γ(E,X) > 1, there are infinitely manypoints of C(K ) whose conjugates in C(Cv ) all belong to Ev , foreach place v of K . If γ(E,X) < 1, there are only finitely manysuch points.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Multi-center Adelic Versions of other ClassicalTheorems

There are multi-center adelic versions, on curves, of the otherclassical applications of capacity theory:

Bilu’s Equidistribution Theorem on Curves: A. Thuillier (Thesis,University of Rennes, 2005);

The Dynamical Equidistribution Theorem for Small Points(Baker-Rumely 2006, Favre-Rivera Letelier 2006,Chambert-Loir, Thuillier, and Autissier 2007)

The Polya-Carlson Theorem on Curves: N. Walters (Thesis,University of Georgia, 2012)

Ferguson’s Theorem on Curves: N. Walters (Thesis, Universityof Georgia, 2012).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Multi-center Adelic Versions of other ClassicalTheorems

There are multi-center adelic versions, on curves, of the otherclassical applications of capacity theory:

Bilu’s Equidistribution Theorem on Curves: A. Thuillier (Thesis,University of Rennes, 2005);

The Dynamical Equidistribution Theorem for Small Points(Baker-Rumely 2006, Favre-Rivera Letelier 2006,Chambert-Loir, Thuillier, and Autissier 2007)

The Polya-Carlson Theorem on Curves: N. Walters (Thesis,University of Georgia, 2012)

Ferguson’s Theorem on Curves: N. Walters (Thesis, Universityof Georgia, 2012).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

New Applications

There are also new applications of capacity theory, usingextremal extremal properties of Green’s functions andequilibrium measures, energy minimization principles, andequidistribution principles, by Igor Pritsker, Paul Fili, ZacharyMiner, Clayton Petsche, Lukas Pottmeyer, and others.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Arithmetic Capacity on Higher Dimensional Varieties

Let K be a global field. For varieties V/K of dimension d ≥ 2,there is a good notion of arithmetic capacity in place:

Chinburg’s Sectional Capacity Sγ(E,D).

Chinburg (1991) has proved a higher dimensional analog ofFekete’s Theorem.

Xinyi Yuan (2008) has given a higher-dimensional analogue ofthe Equidistribution Theorem for small points.

There is as yet no higher dimensional analogue of theFekete-Szegö Theorem or the Pol’ya-Carlson Theorem.However, Bost and Chambert-Loir have obtained results relatedto the Pol’ya-Carlson Theorem, using Arakelov Theory and theMethod of Slopes.

Very recently, Chinburg, Moret-Bailly, Pappas, and Taylor founda higher-dimensional analogue of the Cantor Capacity.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

v-Basic Sets

If v is archimedean and Kv ∼= C, a set Fv ⊂ C(C) is v -basic if itis simply connected, has a piecewise smooth boundary, and isthe closure of its interior.

If v is archimedean and Kv ∼= R, a set Fv ⊂ C(C) is v -basic ifeither

it is simply connected, has a piecewise smooth boundary,and is the closure of its C-interior; orit is contained in C(R) and is homeomorphic to a segment[a,b].

If v is nonarchimedean, a set Fv ⊂ C(Cv ) is v -basic ifit is an open ball B(a, r)− or a closed ball B(a, r); orit is a closed affinoid in the sense of rigid analysis; orfor some separable algebraic extension Lw/Kv (finite orinfinite), it is the intersection of C(Lw ) with an open orclosed ball or an affinoid.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

v-Basic Sets

If v is archimedean and Kv ∼= C, a set Fv ⊂ C(C) is v -basic if itis simply connected, has a piecewise smooth boundary, and isthe closure of its interior.

If v is archimedean and Kv ∼= R, a set Fv ⊂ C(C) is v -basic ifeither

it is simply connected, has a piecewise smooth boundary,and is the closure of its C-interior; orit is contained in C(R) and is homeomorphic to a segment[a,b].

If v is nonarchimedean, a set Fv ⊂ C(Cv ) is v -basic ifit is an open ball B(a, r)− or a closed ball B(a, r); orit is a closed affinoid in the sense of rigid analysis; orfor some separable algebraic extension Lw/Kv (finite orinfinite), it is the intersection of C(Lw ) with an open orclosed ball or an affinoid.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

v-Basic Sets

If v is archimedean and Kv ∼= C, a set Fv ⊂ C(C) is v -basic if itis simply connected, has a piecewise smooth boundary, and isthe closure of its interior.

If v is archimedean and Kv ∼= R, a set Fv ⊂ C(C) is v -basic ifeither

it is simply connected, has a piecewise smooth boundary,and is the closure of its C-interior; orit is contained in C(R) and is homeomorphic to a segment[a,b].

If v is nonarchimedean, a set Fv ⊂ C(Cv ) is v -basic ifit is an open ball B(a, r)− or a closed ball B(a, r); orit is a closed affinoid in the sense of rigid analysis; orfor some separable algebraic extension Lw/Kv (finite orinfinite), it is the intersection of C(Lw ) with an open orclosed ball or an affinoid.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example: A disc with a tail

TheoremLet 0 < R,L ∈ R, and take E∞ = D(0,R) ∪ [R,R + L], a ‘disc

with a tail’. Fix a prime p, and let

Ep = pZ×p ∪ Z×p ∪ p−1Z×p = Qp ∩ (Dp(0,p)\Dp(0,1/p)−) ,

a p-adic annulus. For each prime q 6= p, put Eq = Dq(0,1).Then if

( 34 R+ 1

4R2+RL+L2

R+L ) · p1− 1

p−1 + 1(p−1)2(1+p2+p4) > 1 ,

there are infinitely many algebraic numbers whose whoseconjugates in Cv belong to Ev , for each place v.

If the reverse inequality holds, there are only finitely many.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example: A disc with a tail

The sets in the example are finite unions of ‘basic sets’:

The set E∞ is a union of a set in C which is the closure of itscomplex interior, and a set in R which is the closure of its realinterior. Note that these sets need not be disjoint.

The set Ep is a union of affine translates of Zp:

Ep =1⋃

i=−1

p−1⋃a=1

(a · pi + pi+1Zp

).

The sets Eq = Dq(0,1) for q 6= p are ‘trivial’ with respect to∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example: A disc with a tail

The sets in the example are finite unions of ‘basic sets’:

The set E∞ is a union of a set in C which is the closure of itscomplex interior, and a set in R which is the closure of its realinterior. Note that these sets need not be disjoint.

The set Ep is a union of affine translates of Zp:

Ep =1⋃

i=−1

p−1⋃a=1

(a · pi + pi+1Zp

).

The sets Eq = Dq(0,1) for q 6= p are ‘trivial’ with respect to∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example: A disc with a tail

The sets in the example are finite unions of ‘basic sets’:

The set E∞ is a union of a set in C which is the closure of itscomplex interior, and a set in R which is the closure of its realinterior. Note that these sets need not be disjoint.

The set Ep is a union of affine translates of Zp:

Ep =1⋃

i=−1

p−1⋃a=1

(a · pi + pi+1Zp

).

The sets Eq = Dq(0,1) for q 6= p are ‘trivial’ with respect to∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

Example: A disc with a tail

The sets in the example are finite unions of ‘basic sets’:

The set Ep is a union of affine translates of Zp:

Ep =1⋃

i=−1

p−1⋃a=1

(a · pi + pi+1Zp

).

The sets Eq = Dq(0,1) for q 6= p are ‘trivial’ with respect to∞.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

An Elliptic Curve example

Let E/Q be the elliptic curve y2 = x3 − 256x .The real locus E(R) has two components, with a bounded looplying over the interval [−16,0].

Theorem

There are infinitely many points α ∈ E(Q) whose archimedeanconjugates all belong to the bounded real loop of E(R), whose2-adic conjugates all belong to E(Z2), and whose p-adicconjugates all belong to E(Op) where Op is the ring of integersof Cp

Here X = {o} (the origin of E), and γ(E,X) =∏

v γo(Ev ) where

γo(E∞) = 2, γo(E2) = 2−106/107, and γo(Ep) = 1 for all odd p.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

An Elliptic Curve example

Let E/Q be the elliptic curve y2 = x3 − 256x .The real locus E(R) has two components, with a bounded looplying over the interval [−16,0].

Theorem

There are infinitely many points α ∈ E(Q) whose archimedeanconjugates all belong to the bounded real loop of E(R), whose2-adic conjugates all belong to E(Z2), and whose p-adicconjugates all belong to E(Op) where Op is the ring of integersof Cp

Here X = {o} (the origin of E), and γ(E,X) =∏

v γo(Ev ) where

γo(E∞) = 2, γo(E2) = 2−106/107, and γo(Ep) = 1 for all odd p.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

A Fermat Curve example

Take K = Q and consider the Fermat Curve F with affineequation xp + yp = 1.

It has p points at∞; let X be that set of points.

Take 0 < R ∈ R and put E∞ = {(x , y) ∈ F(C) : |x | ≤ R}.At the prime p, let Lw/Qp be the extension Lw = Qp(ζp).Put Ep = F(OLw ).For all other primes q, let Eq be the X-trivial set.

McCallum has determined a regular model for F over OLw ; ithas np components of a certain type, corresponding to thenumber of nontrivial linear Fp-rational factors of the equation((x − y)p − (xp − yp))/p ≡ 0 (mod p).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

A Fermat Curve example

Take K = Q and consider the Fermat Curve F with affineequation xp + yp = 1.

It has p points at∞; let X be that set of points.

Take 0 < R ∈ R and put E∞ = {(x , y) ∈ F(C) : |x | ≤ R}.At the prime p, let Lw/Qp be the extension Lw = Qp(ζp).Put Ep = F(OLw ).For all other primes q, let Eq be the X-trivial set.

McCallum has determined a regular model for F over OLw ; ithas np components of a certain type, corresponding to thenumber of nontrivial linear Fp-rational factors of the equation((x − y)p − (xp − yp))/p ≡ 0 (mod p).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

A Fermat Curve example

Take K = Q and consider the Fermat Curve F with affineequation xp + yp = 1.

It has p points at∞; let X be that set of points.

Take 0 < R ∈ R and put E∞ = {(x , y) ∈ F(C) : |x | ≤ R}.At the prime p, let Lw/Qp be the extension Lw = Qp(ζp).Put Ep = F(OLw ).For all other primes q, let Eq be the X-trivial set.

McCallum has determined a regular model for F over OLw ; ithas np components of a certain type, corresponding to thenumber of nontrivial linear Fp-rational factors of the equation((x − y)p − (xp − yp))/p ≡ 0 (mod p).

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

A Fermat Curve example

Take K = Q and consider the Fermat Curve F with affineequation xp + yp = 1.

It has p points at∞; let X be that set of points.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory

A Fermat Curve example

Theorem

There are infinitely many points of F(Q) which have all theirconjugates in Ev for each v if

R · p− p(2p−1)

(p−1)2((2np+2)p−np) > 1,

and only finitely many if the opposite inequality holds.

Robert Rumely A Survey of Arithmetic Applications of Capacity Theory