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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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The Euler binary partition function and subdivision schemes
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by Vladimir Yu. Protasov PDF
Math. Comp. 86 (2017), 1499-1524 Request permission


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
  • Email:
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1499-1524
  • MSC (2010): Primary 05A17, 39A99, 11P99, 65D17, 15A60
  • DOI:
  • MathSciNet review: 3614025