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Mathematics of Computation

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The Euler binary partition function and subdivision schemes

Author: Vladimir Yu. Protasov
Journal: Math. Comp. 86 (2017), 1499-1524
MSC (2010): Primary 05A17, 39A99, 11P99, 65D17, 15A60
Published electronically: July 26, 2016
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Abstract: 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 behaviour of $ b(k)$ as $ k \to \infty $ 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.

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Vladimir Yu. Protasov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, and Faculty of Computer Science of National Research University Higher School of Economics, Moscow, Russia

Keywords: Binary partition function, asymptotics, subdivision scheme, refinement equation, binary expansion, positive matrix, cyclotomic polynomial
Received by editor(s): July 24, 2015
Received by editor(s) in revised form: October 15, 2015
Published electronically: July 26, 2016
Additional Notes: The work was supported by RFBR grants nos. 14-01-00332 and 16-04-00832, and by a grant of the Dynasty Foundation
Article copyright: © Copyright 2016 American Mathematical Society

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