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## The Euler binary partition function and subdivision schemes

Mathematics of Computation. 2017. Vol. 86. No. 305. P. 1499-1524.

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