Counting Non-Standard Binary Representations


Let A be a finite subset of N including 0 and let fA(n) be the number of ways to write n = ∑∞i=0 ǫi2 , where ǫi ∈ A. We consider asymptotics of the summatory function sA(r,m) of fA(n) from m2 r to m2 − 1, and show that sA(r,m) ∼ c(A,m) ∣A∣ r for some nonzero c(A,m) ∈ Q. 


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