Note on a conjecture of S\'ark\"ozy on special sequences

Abstract

Let $\alpha>1$ be an irrational number and $k\ge 2$ a positive integer. Let $f(x)$ be a polynomial with positive integer coefficients. Solving a 2001 problem of S\'ark\"ozy on special sequences, Hegyv\'ari proved in 2003 that there exists an infinite sequence $A$ with density $\frac{1}{k}-\frac{1}{k\alpha}$ such that $$ \big\{f(a_1)+\ldots+f(a_k): a_i\in A, 1\le i\le k\big\}\cap \big\{\lfloor n\alpha\rfloor: n\in \mathbb{N}\big\}=\emptyset. $$ Hegyv\'ari also proved that the density given by him is optimal for $k=2$. In this article, we show that the density $\frac{1}{k}-\frac{1}{k\alpha}$ given by Hegyv\'ari is actually optimal for all $k\ge 2$.