A modular version of the Brunn-Minkowski inequality and its applications

Let $α>1$ be an irrational number and 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 such that (a₁)+…+f(aₖ): aᵢ∈ A, 1≤ i≤ k\∩ \ nα: n∈ N\=∅. Hegyvári also proved that the density given by him is optimal for $k=2$. In this article, we show that the density 1/k-1/kα given by Hegyvári is actually optimal for all k≥ 2. The proof will follow from a modular version of the Brunn-Minkowski inequality established here.