REPRESENTATION OF A 2-POWER AS SUM OF k 2-POWERS: THE ASYMPTOTIC BEHAVIOR

Abstract

A k-representation of an integer ℓ is a representation of ℓ as sum of k powers of 2, where representations differing by the order are considered as distinct. Let [Formula: see text] be the maximum number of such representations for integers ℓ whose binary representation has exactly σ non-zero digits. [Formula: see text] can be recovered from [Formula: see text] via an explicit formula, thus in some sense [Formula: see text] is the fundamental object. In this paper we prove that [Formula: see text] tends to a computable limit as k diverges. This result improves previous bounds which were obtained with purely combinatorial tools.