Counting non-standard binary representations

Abstract

Let A be a finite subset of N including 0 and f A (n) be the number of ways to write n = ∞ i=0 ǫ i 2 i , where ǫ i ∈ A. We consider asymptotics of the summatory function s A (r, m) of f A (n) from m2 r to m2 r+1 − 1 and show that s A (r, m) ≈ c(A, m) |A| r for some c(A, m) ∈ Q.