Note on a conjecture of Sárközy on special sequences

Abstract Let $$\alpha >1$$ α > 1 be an irrational number and $$k\ge 2$$ k ≥ 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 $$\frac{1}{k}-\frac{1}{k\alpha }$$ 1 k - 1 k α such that $$ \big \{f(a_1)+\ldots +f(a_k): a_i\in A, 1\le i\le k\big \}\cap \big \{\lfloor n\alpha \rfloor : n\in \mathbb {N}\big \}=\emptyset . $$ { f ( a 1 ) + … + f ( a k ) : a i ∈ A , 1 ≤ i ≤ k } ∩ { ⌊ n α ⌋ : n ∈ N } = ∅ . Hegyvári also proved that the density given by him is optimal for $$k=2$$ k = 2 . In this article, we show that the density $$\frac{1}{k}-\frac{1}{k\alpha }$$ 1 k - 1 k α given by Hegyvári is actually optimal for all $$k\ge 2$$ k ≥ 2 .