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

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 $R$.