Translation-invariant equations in $\mathbb{F}_p^n$ and groups

Abstract

In this paper, we explore how large a subset of a finite group can be without containing non-trivial solutions to linear equations of the form $aX+bY=cZ+dU$ (where $a+b=c+d$). These equations naturally generalize the classic concept of Sidon sets. Using Tao's slice rank method, we first establish strict upper bounds for the size of such solution-free sets in the vector space $\mathbb{F}_{p}^{n}$. Next, we turn to cyclic groups $\mathbb{Z}_{m}$ and construct surprisingly large sets that have no solutions for the asymmetric equation $X+5Y=3U+3Z$. These constructions achieve a logarithmic density of roughly $0.5283$, breaking the expected $0.5$ barrier for any sufficiently large modulus $m$. Finally, we show that this high-density behavior extends to a wider family of equations whose coefficients follow simple rules modulo $8$.