ON A SIMPLE PROOF OF SLIGHTLY CURVED SEQUENCES CONTAINING ARBITRARILY LONG ARITHMETIC PROGRESSIONS (Analytic Number Theory and Related Topics)

Abstract

The author and Yoshida proved that a strictly increasing sequence {a(n)}n∈A of positive integers, which can be written as a(n) = f(n) + O(1) for some function f : R → R satisfying f"(x) = O(1/xα) for some a > 0, must contain arbitrarily long arithmetic progressions for all A C N with positive upper Banach density. In this article, we get a simple proof and the same conclusion ifwe replace the condition f"(x) = O(1/xα) to f"(x) = o(1).