Cyclic permutations of large subsets with polynomial values in multiplicative subgroups of finite fields

Abstract

Let $f(t)\in \mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge 2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv 1(\mathrm{mod}k)$, we study cyclic orderings of subsets $A\subseteq {\mathbb{F}}_p$ for which $f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on ${\mathbb{F}}_p$, and spectral graph methods, we prove that every $A$ with $\#A\ge p+1-\lfloor N(p,k,f)/2\rfloor$ admits such an ordering, where $N(p,k,f)=\#\{x\in {\mathbb{F}}_p:f(x)\in D_k({\mathbb{F}}_p)\}.$ We also give lower and upper bounds for the optimal universal threshold.

21 pages. Comments are welcome