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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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

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
  • Email: v-protassov@yandex.ru
  • 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: https://doi.org/10.1090/mcom/3128
  • MathSciNet review: 3614025