On block Behrend sequences

Abstract

Let A be a strictly increasing sequence of integers exceeding 1 and letdenote its set of multiples. We say that A is a Behrend sequence if M(A) has asymptotic density 1. The theory of sets of multiples was first developed in the late thirties, under the influence of Erdős, Besicovitch, and others. An account of the classical notions is presented in the book of Halberstam and Roth (1966), and recent progress in the area may be found in Hall and Tenenbaum [7], Erdős and Tenenbaum [4], Ruzsa and Tenenbaum [9]. As underlined by Erdős in [3], one of the central problems in the field is that of finding general criteria to decide whether a given sequence A is Behrend.