A note on Behrend sequences

Abstract

The concept of a Behrend sequence is one of the most fundamental and challenging in the theory of sets of multiples. A sequence A of integers exceeding 1 is called a Behrend sequence if almost all integers n have at least one divisor in A, or, in other words, if its set of multiples M(A) = {ma : m 1, a ∈ A} has natural density 1. This was recently defined formally by Hall (1990), but the idea has been constantly used by Erdős in the last half-century. Recent progress on this topic may be found in Hall-Tenenbaum (1992), Erdős-Hall-Tenenbaum (1994), Tenenbaum (1994). By the Davenport-Erdős theorem (1937, 1951), any set of multiples M(A) has a logarithmic density δM(A), equal to its lower asymptotic density, moreover