A proof of a conjecture of Lev

Abstract

We say that a set of the form [Formula: see text] for some [Formula: see text] is an interval. For a nonempty finite subset [Formula: see text] of [Formula: see text] and [Formula: see text], Vsevolod Lev proved in [Optimal representations by sumsets and subset sums, J. Number Theory 62(1) (1997) 127–143] some results about the existence of long intervals contained in the [Formula: see text]-fold iterated sumset of [Formula: see text]. Furthermore, in the same paper, he proposed a conjecture [Optimal representations by sumsets and subset sums, J. Number Theory 62(1) (1997) 127–143, Conjecture 1], see also [V. Lev, Consecutive integers in high-multiplicity sumsets, Acta Math. Hungar. 129(3) (2010) 245–253, Conjecture 1]. Lev proved some particular cases of his conjecture, and he showed that these few cases have important applications. In this paper, we prove his conjecture.