The Euler binary partition function and subdivision schemes

For an arbitrary set D of nonnegative integers, we consider the Euler binary partition function b(k) which equals the total number of binary expansions of an integer k with “digits” from D. By applying the theory of subdivision schemes and refinement equations, the asymptotic behavior of b(k) as k → ∞ is characterized. For all finite D, we compute the lower and upper exponents of growth of b(k), find when they coincide, and present a sharp asymptotic formula for b(k) in that case, which is done in terms of the corresponding refinable function. It is shown that b(k) always has a constant exponent of growth on a set of integers of density one. The sets D for which b(k) has a regular power growth are classified in terms of cyclotomic polynomials.