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Sum-difference exponents for boundedly many slopes, and rational complexity

Terence Tao

TL;DR

The paper investigates how the sum-difference exponent ${\operatorname{SD}}(R;s)$, which governs entropy-based bounds on projections and underpins Kakeya-type dimension estimates, behaves when the slope set $R$ is fixed and the target slope $s$ varies. It introduces rational complexity $D(R;s)$ to quantify how efficiently $s$ can be expressed in terms of $R$, and proves that ${\operatorname{SD}}(R;s)$ approaches 2 with a rate controlled by $D(R;s)$ (and related quantities) via a detailed entropic framework. The core contribution is a reduction to an independent-case setting, accomplished through a sequence of steps that produce good configurations, invertible projections, and controlled doubling, culminating in a bound involving the function $f(R;s)$. The results yield precise asymptotics in the single-slope case and general bounds for arbitrary finite $R$, and they illuminate a continuous-limit regime where $s$ approaches a real parameter $\alpha$, characterized by a variational constant $c_\alpha$. This work thus links rational-expression complexity to entropy-based sumset estimates and Kakeya-type dimension bounds, providing a structured path to tighten the arithmetic Kakeya conjecture in the regime of bounded slopes.

Abstract

The dimension of Kakeya sets can be bounded using sum-difference exponents $\SD(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions, asserts that the infimum of such exponents is $1$. The best upper bound on this infimum currently is $1.67513\dots$. In this note, inspired by numerical explorations from the tool \texttt{AlphaEvolve}, we study the regime where the cardinality of the set of slopes $R$ is bounded. In this regime, we establish that these exponents converge to $2$ at a rate controlled by the \emph{rational complexity} of $s$ relative to $R$, which measures how efficiently $s$ can be expressed as a rational combination of slopes in $R$.

Sum-difference exponents for boundedly many slopes, and rational complexity

TL;DR

The paper investigates how the sum-difference exponent , which governs entropy-based bounds on projections and underpins Kakeya-type dimension estimates, behaves when the slope set is fixed and the target slope varies. It introduces rational complexity to quantify how efficiently can be expressed in terms of , and proves that approaches 2 with a rate controlled by (and related quantities) via a detailed entropic framework. The core contribution is a reduction to an independent-case setting, accomplished through a sequence of steps that produce good configurations, invertible projections, and controlled doubling, culminating in a bound involving the function . The results yield precise asymptotics in the single-slope case and general bounds for arbitrary finite , and they illuminate a continuous-limit regime where approaches a real parameter , characterized by a variational constant . This work thus links rational-expression complexity to entropy-based sumset estimates and Kakeya-type dimension bounds, providing a structured path to tighten the arithmetic Kakeya conjecture in the regime of bounded slopes.

Abstract

The dimension of Kakeya sets can be bounded using sum-difference exponents for various sets of rational slopes and output slope ; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions, asserts that the infimum of such exponents is . The best upper bound on this infimum currently is . In this note, inspired by numerical explorations from the tool \texttt{AlphaEvolve}, we study the regime where the cardinality of the set of slopes is bounded. In this regime, we establish that these exponents converge to at a rate controlled by the \emph{rational complexity} of relative to , which measures how efficiently can be expressed as a rational combination of slopes in .

Paper Structure

This paper contains 15 sections, 13 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The arithmetic Kakeya conjecture
  3. Asymptotic behavior
  4. Notation
  5. Acknowledgments
  6. A review of entropy sum set estimates
  7. A reduction to the independent case
  8. First step: obtaining a good configuration
  9. Second step: ensuring an invertible projection
  10. Third step: controlling one doubling constant
  11. Fourth step: controlling a second doubling constant
  12. Fifth step: controlling projections of conditionally independent variables
  13. Final step: using the quantity $f(R;s)$
  14. Proof of Theorem \ref{['main-2']}
  15. A continuous limit

Key Result

Theorem 1.3

Let $R$ be a finite family of slopes containing $0,1,\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.

Figures (1)

  • Figure 1: Lower bounds on ${\operatorname{SD}}(\{0,1,\infty\}; s)$ obtained by AlphaEvolve for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).

Theorems & Definitions (25)

  • Definition 1.1: Rational complexity
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4: Continuous limit
  • Proposition 2.1
  • proof
  • Lemma 2.2: Entropic BSG
  • proof
  • Definition 3.1: Independent sum-difference constant
  • Theorem 3.2: Reduction to independent case
  • ...and 15 more