Judicious partitions for restricted self-sumsets in cyclic groups

Abstract

We study the minimax problem for restricted two-fold self-sumsets in $k$-colorings of $\mathbb{Z}_n$. For primes $p$ with $2\le k\le p$ we determine the exact minimum $\max\{0,\,2\lceil p/k\rceil-3\}$. For general $n$ (with $m=\lceil n/k\rceil$) we bound the optimum between a size term $\min\{p(n),\,2m-3\}$ and a periodicity term $f\big(n/q(n,k)\big)$, and show these bounds are tight when $2m-3\le p(n)$ or $f\big(n/q(n,k)\big)\le \min\{p(n),\,2m-3\}$. We further prove a stability inequality and a threshold theorem that force concentration in a single subgroup coset near the periodic scale. In the prime case with $m\ge 5$ and $2m-3<p$, every optimal coloring contains a class of size $m$ that is an arc (an arithmetic progression up to an affine automorphism). Our approach combines the restricted Erd\H{o}s--Heilbronn phenomenon with block/coset colorings and an injectivity window.