Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

Unconventional height functions in simultaneous Diophantine approximation David Simmons Ohio State University

Bibliography Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

L. Fishman and D. S. Simmons, Unconventional height functions in simultaneous Diophantine approximation, http://arxiv.org/abs/1401.8266, 2014, preprint. L. Fishman, D. S. Simmons, and M. Urba´ nski, Diophantine approximation in Banach spaces, http://arxiv.org/abs/1302.2275, 2013, to appear J. Th´eor. Nombres Bordeaux.

Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

1 Introduction 2 Main results

Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

3 Proof sketches

Classical simultaneous Diophantine approximation Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

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In classical one-dimensional Diophantine approximation, the height of a rational number p/q is H0 (p/q) = q, assuming that p/q is in reduced form, i.e. gcd(p, q) = 1. This is generalized to higher dimensions as follows: the standard height of a rational vector p/q ∈ Qd is   p1 pd Hstd ,..., = q, q q assuming that gcd(p1 , . . . , pd , q) = 1. Equivalently,   p1 pd Hstd ,..., = lcm(q1 , . . . , qd ), q1 qd assuming that each fraction pi /qi is in reduced form.

(LCM)

Nonstandard height functions Unconventional height functions in Diophantine approximation David Simmons Introduction



 p1 pd Hstd ,..., = lcm(q1 , . . . , qd ), (LCM) q1 qd The formula (LCM) begs the question: Why lcm? There are plenty of functions which input a d-tuple of integers and output a single integer, for example

Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

max, min, prod. Definition Fix d ≥ 1, and for each function Θ : Nd → N let   p1 pd HΘ ,..., = Θ(q1 , . . . , qd ), q1 qd where we assume that p1 /q1 , . . . , pd /qd are in reduced form.

Philosophical differences Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

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The standard height function is natural if you look at the pair (Rd , Qd ) and say “I have a vector space with a lattice in it, and I care about all points of the form p/q, where p is in my lattice and q ∈ N.” The height functions Hmax , Hmin , and Hprod are natural if you look at the pair (Rd , Qd ) and say “I just took the dth Cartesian power of the pair (R, Q).” These philosophical differences translate into different methods used to analyze the different height functions. The standard height function has connections to homogeneous dynamics on the space of all lattices in Rd . By contrast, study of the height functions Hmax , Hmin , Hprod must take a more “coordinate-wise” approach.

A reformulation of Dirichlet’s theorem Unconventional height functions in Diophantine approximation David Simmons Introduction

Theorem (Dirichlet’s Theorem (weak version)) For every x ∈ Rd \ Qd ,



1 p

x − p ≤ for infinitely many ∈ Qd .

1+1/d q q q

Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Equivalently, kx − rn k ≤ ψ1+1/d ◦ Hlcm (rn ) for some sequence Qd 3 rn → x. (DIR)

Proof sketches

Here we use the notation ψα (q) =

1 · qα

Motivating questions Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

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Formula (DIR) motivates the following questions: Given d ≥ 1 and Θ ∈ {max, min, prod}, Question (Exponents of irrationality) What is the largest α ≥ 0 such that for all x ∈ Rd \ Qd , kx − rn k ≤ ψα ◦ HΘ (rn ) for some sequence Qd 3 rn → x?

Question (Dirichlet functions) What is the fastest decaying function ψ such that for all x ∈ Rd \ Qd , kx − rn k ≤ ψ ◦ HΘ (rn ) for some sequence Qd 3 rn → x?

Exponents of irrationality: definition Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Definition The largest α ≥ 0 such that for all x ∈ Rd \ Qd , kx − rn k ≤ ψα ◦ HΘ (rn ) for some sequence Qd 3 rn → x is called the exponent of irrationality of the height function HΘ . It will be denoted ωd (HΘ ). Note that according to this definition,

Proof sketches

ωd (Hlcm ) = 1 + 1/d.

Exponents of irrationality of nonstandard height functions Unconventional height functions in Diophantine approximation David Simmons

We may now state our first theorem: Theorem (Fishman, S. 2014) For any d ≥ 1,

Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

d (d − 1)(d−1)/d ωd (Hmin ) = 2 2 ωd (Hprod ) = · d ωd (Hmax ) =

if d ≥ 2

We can see already that the height function Hmax is the most “interesting” of the three.

Comparison with a priori inequalities Unconventional height functions in Diophantine approximation David Simmons Introduction

Remark: The inequalities min ≤ prod1/d ≤ max ≤ lcm ≤ prod automatically imply that ωd (Hprod ) ≤ ωd (Hlcm ) ≤ ωd (Hmax ) ≤ dωd (Hprod ) ≤ ωd (Hmin ).

Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

i.e. 2/d ≤ 1 + 1/d ≤

d ≤2≤2 (d − 1)(d−1)/d

When d ≥ 3, all inequalities are strict except the last. When d = 2, the third inequality is an equality.

Dirichlet functions Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

The second question is more subtle. Definition Given d ≥ 1 and Θ ∈ {max, min, prod}, a function ψ : N → (0, ∞) is called Dirichlet on Rd with respect to HΘ if for every x ∈ Rd \ Qd , there exists Cx > 0 such that kx − rn k ≤ Cx ψ ◦ HΘ (rn ) for some sequence Qd 3 rn → x. If the constant Cx can be made arbitrarily small, then the point x is called well approximable with respect to ψ.

Optimally Dirichlet functions Unconventional height functions in Diophantine approximation David Simmons Introduction

Theorem (Fishman, S., Urbanski 2013) If ψ is a Dirichlet function, then the following are equivalent: Every x ∈ Rd \ Qd is well approximable with respect to ψ. There exists a Dirichlet function φ : N → (0, ∞) such that

Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

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φ(q) −−−→ 0. ψ(q) q→∞ If these conditions do not hold, we say that the Dirichlet function ψ is optimally Dirichlet. It is well-known that ψ1+1/d is optimally Dirichlet on Rd with respect to Hlcm .

A division into cases Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

Theorem (Fishman, S. 2014) For d ≥ 1, ψ2 is optimally Dirichlet on Rd with respect to Hmin ψ2/d is optimally Dirichlet on Rd with respect to Hprod ψ2 is optimally Dirichlet on R2 with respect to Hmax .

Question If d ≥ 3, is there an optimally Dirichlet function on Rd with respect to Hmax ?

Hardy L-functions Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Definition A Hardy L-function is a function which can be expressed using only the elementary arithmetic operations +, −, ×, ÷, exponents, logarithms, and real-valued constants, and which is well-defined on some interval of the form (t0 , ∞). For example, for any C , α ≥ 0 the function ψ(q) = q −α+C / log

2

log(q)

is a Hardy L-function.

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Remark The class of Hardy L-functions includes almost all functions that one naturally encounters in dealing with “analysis at infinity”, except for those with oscillatory behavior.

No optimal Dirichlet function Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

Theorem (Fishman, S. 2014) Suppose d ≥ 3. Then no Hardy L-function is optimally Dirichlet on Rd with respect to the height function Hmax . Thus there is some sort of “unreachable boundary” between the class of Dirichlet functions and the class of non-Dirichlet functions. This situation is not unprecedented in Diophantine approximation: a famous example is given by Theorem (Khinchin’s theorem) Given a decreasing function ψ : N → (0, ∞), the following are equivalent: For Lebesgue a.e. x ∈ R, there exist infinitely many p/q ∈ Q such that |x − p/q| ≤ ψ(q). P The series ∞ q=1 ψ(q) diverges.

A well-known result Unconventional height functions in Diophantine approximation David Simmons

The boundary between when a series converges and when it diverges is well-known. Theorem For each N ≥ −1 and α ≥ 0 let

Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

ψN,α (q) =

N Y

!

1

1

!α

log(N+1) (q) 1  α . = (N) q log(q) · · · log (q) log(N+1) (q) i=0

log(i) (q)

Proof sketches

Then the series Here

P∞

q=1 ψN,α (q)

diverges if and only if α ≤ 1.

log(i) (q) = log · · · log(q). | {z } i times

Back to the question at hand Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

We would like similar information about a boundary between functions ψ which are Dirichlet on Rd with respect to Hmax and those which are not. Notation Fix d ≥ 3. Let Dd denote the class of all functions which are Dirichlet on Rd with respect to Hmax , and let γd = (d − 1)1/d > 1 −(d−1)

αd = ωd (Hmax ) = γd + γd

=

d · (d − 1)(d−1)/d

A first step Unconventional height functions in Diophantine approximation David Simmons

Theorem (Fishman, S. 2014) For each C > 0 let ψ(q) = q −αd +C / log

Introduction

2

log(q)

.

Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Then ψ ∈ Dd if and only if C > C0 :=

Proof sketches

Example ψαd ∈ / Dd .

dγd log2 (γd ) · 8

Adding more error terms Unconventional height functions in Diophantine approximation David Simmons

For each N ≥ 1 and C ≥ 0, let fN,C (q) =

Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

N Y n X

1

n=2 i=2

log(i) (q)

!2 +C

N+1 Y

1

i=2

log(i) (q)

!2

and let ψN,C (q) = q −αd +C0 fN,C (q) .

Theorem (Fishman, S. 2014)

Proof sketches

ψN,C ∈ Dd if and only if C > 1. The theorem on the previous slide is precisely the special case N = 1.

Back to Hardy L-functions Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

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Theorem (Corollary of a result of G. H. Hardy) If ψ is a Hardy L-function, then there exist N, C such that either ψ  ψN,C ∈ Dd or ψ ≺ ψN,C ∈ / Dd . Thus, the theorem of the previous slide provides a complete answer as to whether or not ψ ∈ Dd in the case where ψ is a Hardy L-function.

Continued fractions Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

Fix Θ ∈ {max, min, prod}, a point x = (x1 , . . . , xd ) ∈ Rd , and a Hardy L-function ψ. We want to answer the following question: Does there exist Cx > 0 such that kx − rn k ≤ Cx ψ ◦ HΘ (rn ) for some sequence Qd 3 rn → x? (i)

(i)

For each i = 1, . . . , d, let (pn /qn )∞ n=1 be the convergents of the continued fraction expansion of xi . Since best approximations are always convergents, we’ll mostly be interested in rational numbers of the form ! (1) (d) pn1 pnd r= , · · · , (d) . (1) qn1 qnd For such r, we have d

kx − rk  max

1

i=1 q (i) q (i) ni ni +1

·

Reformulating the question Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

So the question on the previous slide can be reformulated: Do we have   1 d (1) (d) max (i) (i) ≤ Cx ψ Θ(qn1 , . . . , qnd ) i=1 q q ni ni +1 for infinitely many d-tuples (ni )i ∈ Nd ? We notice that this (i) d question depends only on the sequences ((qn )∞ n=1 )i=1 rather than on the point x. (i) d Remark: Given any d-tuple ((qn )∞ n=1 )i=1 of “geometrically increasing” sequences, there exists x whose d-tuple of sequences of denominators of convergents is asymptotically (i) d equal to ((qn )∞ n=1 )i=1 .

Reformulating the question Unconventional height functions in Diophantine approximation

Now let’s specialize to the case Θ = max. Write ψ(q) = e −Ψ(log(q))

David Simmons Introduction

(i)

(i)

for some function Ψ, and write bn = log(qn ). Then

Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

d

1

i=1

qni qni +1

max

(i) (i)

  (1) (d) ≤ ψ max(qn1 , . . . , qnd )

if and only if

Proof sketches

  d (i) (i) (1) (d) min[bni + bni +1 ] ≥ Ψ max(bn1 , . . . , bnd ) . i=1

A simplification Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

Suppose that the answer to our question is no, i.e. that   d (i) (i) (1) (d) min[bni + bni +1 ] ≤ Ψ max(bn1 , . . . , bnd ) i=1

(NO)

for all but finitely many d-tuples (ni )i ∈ Nd . Then (major step in the proof) we may without loss of generality assume that (i)

bn = Adn+i for some increasing sequence (Ak )∞ k=1 . Fix k ∈ N, and let ni be maximal satisfying dni + i ≤ k. Substituting into (NO) gives Ak−d+1 + Ak+1 ≤ Ψ(Ak ).

(NOv2)

The equation (NOv2) Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Ak−d+1 + Ak+1 ≤ Ψ(Ak ).

(NOv2)

Although deceptively simple, it is quite complicated to determine for which functions Ψ the equation (NOv2) has a solution. To finish this talk, let’s consider the simple case where Ψ(b) = αb for some α ≥ 0 (corresponding to ψ(q) = q −α ). In this case, one can easily guess a solution to (NOv2); namely, the solution Ak = γ k , γ > 1. (GEOM)

Proof sketches

(GEOM) is a solution to (NOv2) if and only if γ + γ −(d−1) ≤ α.

Motivation for the values γd , αd Unconventional height functions in Diophantine approximation David Simmons Introduction Main results Dirichlet’s theorem Exponents of irrationality Dirichlet functions The case d ≥ 3, H = Hmax

Proof sketches

The inequality γ + γ −(d−1) ≤ α provides motivation for the values γd = (d − 1)1/d −(d−1)

αd = γd + γd

=

d (d − 1)(d−1)/d

introduced earlier. Specifically, the function f (γ) = γ + γ −(d−1) achieves its unique minimum at γ = γd , and the value at that point is f (γd ) = αd .