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Sums of Reciprocals of Fractional Parts II

Reynold Fregoli

TL;DR

The paper develops a general lattice-point counting framework for non-convex regions arising in Diophantine approximation by introducing φ-multiplicatively-badly-approximable matrices and proving a sharp upper bound for the lattice intersection count M(L,ε,R,T) that depends only on the geometric mean of the T-components. This geometric-mean bound, combined with tessellation techniques and a careful analysis of successive minima under diagonal maps, yields unconditional almost-sure asymptotics for sums of reciprocals of fractional parts: S(α,T) ≍ log T̄ · log T1 · ... · log Tn and S*(α,T) ≍ (log T)^{n+1} for a.e. α. The results settle conjectures by Beresnevich–Haynes–Velani and extend lower-bound frameworks to sharp upper bounds in higher dimensions, including non-averaged sums with general φ. Methodologically, the work fuses geometry-of-numbers tessellations with Schmidt–Cassels-type probabilistic arguments, producing a versatile approach to non-convex lattice-point problems with Diophantine applications.

Abstract

We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of magnitude of sums of reciprocals of fractional parts and solve a conjecture posed by Beresnevich, Haynes, and Velani. The methods are based both on the geometry of numbers and on probability theory.

Sums of Reciprocals of Fractional Parts II

TL;DR

The paper develops a general lattice-point counting framework for non-convex regions arising in Diophantine approximation by introducing φ-multiplicatively-badly-approximable matrices and proving a sharp upper bound for the lattice intersection count M(L,ε,R,T) that depends only on the geometric mean of the T-components. This geometric-mean bound, combined with tessellation techniques and a careful analysis of successive minima under diagonal maps, yields unconditional almost-sure asymptotics for sums of reciprocals of fractional parts: S(α,T) ≍ log T̄ · log T1 · ... · log Tn and S*(α,T) ≍ (log T)^{n+1} for a.e. α. The results settle conjectures by Beresnevich–Haynes–Velani and extend lower-bound frameworks to sharp upper bounds in higher dimensions, including non-averaged sums with general φ. Methodologically, the work fuses geometry-of-numbers tessellations with Schmidt–Cassels-type probabilistic arguments, producing a versatile approach to non-convex lattice-point problems with Diophantine applications.

Abstract

We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of magnitude of sums of reciprocals of fractional parts and solve a conjecture posed by Beresnevich, Haynes, and Velani. The methods are based both on the geometry of numbers and on probability theory.
Paper Structure (23 sections, 35 theorems, 229 equations)

This paper contains 23 sections, 35 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Counting Points and Diophantine Properties of Lattices
  3. Sums of Reciprocals of Fractional Parts
  4. Methods
  5. Tessellations
  6. Proof of Theorem \ref{['prop:cor1']}
  7. Proof of Proposition \ref{['prop:firstminima']}
  8. Construction of a Basis
  9. Proof
  10. Counting Lattice Points
  11. Proof of Lemma \ref{['lem:quickcase']}
  12. Proof of Lemma \ref{['lem:slowcase']}
  13. Proof of Lemma \ref{['lem:exponentswelldef']}
  14. Proof of Theorem \ref{['thm:BHV']}
  15. Estimating the sums $S_{1}$ and $S_{1}^{*}$
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $\boldsymbol{L}\in\mathbb{R}^{m\times n}$ be a $\phi$-multiplicatively badly approximable matrix and suppose that $R^{m}/\varepsilon\geq e^{m}$, where $e=2.71828\dots$ is the base of the natural logarithm. Then for all $\boldsymbol T\in[1,+\infty)^{n}$ we have that where $\bar{T}:=\left(T_{1}\dotsm T_{n}\right)^{1/n}$.

Theorems & Definitions (54)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 44 more