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

TL;DR

This paper investigates sum-difference exponents related to Kakeya sets with rational slopes, showing they approach 2 at a rate determined by the rational complexity of the slopes, providing insights into the arithmetic Kakeya conjecture.

Contribution

It establishes that sum-difference exponents converge to 2 when the number of slopes is bounded, with the convergence rate linked to the rational complexity of the slopes.

Findings

Exponents approach 2 as the number of slopes is bounded.

Convergence rate depends on the rational complexity of the slopes.

Provides a new perspective on the arithmetic Kakeya conjecture.

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 $R$.