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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
MathSciNet review: 3614025
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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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Additional Information

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
MR Author ID: 607472

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