Note on a conjecture of Sárközy on special sequences

Let α >1 α > 1 be an irrational number and k≥ 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 1/k-1/kα 1 k - 1 k α such that (a₁)+… +f(aₖ): aᵢ∈ A, 1≤ i≤ k \∩ \ nα : n∈ N \=∅ . { 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 1/k-1/kα 1 k - 1 k α given by Hegyvári is actually optimal for all k≥ 2 k ≥ 2 .