On the number of optimal base 2 representations of integers

We study representations of integers n in binary expansions using the digits 0, ±1. We analyze the average number of such representations of minimal “weight” ( = number of non-zero digits). The asymptotic main term of this average involves a periodically oscillating function, which is analyzed in some detail. The main tool is the construction of a measure on [−1,1], which encodes the number of representations