Note on a conjecture of Sárközy 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 Sárközy on special sequences, Hegyvári proved in 2003 that there exists an infinite sequence $A$ with density $\frac{1}{k}-\frac{1}{k\alpha }$ such that $$\{f(a_1)+\dots +f(a_k):a_i\in A,1\le i\le k\}\cap \{\lfloor n\alpha \rfloor :n\in \mathbb{N}\}=\varnothing .$$ Hegyvári 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 Hegyvári is actually optimal for all $k\ge 2$.

The proof of Lemma 2 (Theorem 1.2 in the first version) was significantly simplified