Gelfond’s problem for expansions in linear recurrent bases
https://doi.org/10.22405/2226-8383-2025-26-5-110-136
Abstract
Gelfond obtained a result on the uniform distribution of sums of digits of 𝑏-ary expansions
of natural numbers over residue classes modulo 𝑑 for an arbitrary 𝑑. Later, Lamberger and
Thuswaldner, using deep estimates of trigonometric sums, obtained an analogue of Gelfond’s theorem, in which instead of 𝑏-ary expansions, expansions over linear recurrent bases satisfying the Parry condition and some additional condition on the coefficients, are used. In this paper, we give a new, simpler and self-contained, proof of the Lamberger-Tkuswaldner theorem. Our proof is purely combinatorial and require only Parry condition. In addition, we give a quite simple explicit formula for the exponent in the remainder term. In contrast to the Lamberger-Thuswaldner result, obtained exponent depends only on 𝑑 and the order of the linear recurrent sequence, but not on its coefficients. However, our result does not include the equidistribution of the sums of the digits modulo 𝑑 of natural numbers running from an arbitrary arithmetic progression, which was also proved by Lamberger and Thuswaldner.
At the end of the paper, some unsolved problems are briefly discussed.
About the Authors
Alla Adolfovna Zhukova
Russian Presidential Academy of National Economy and Public Administration (Vladimir branch)
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Anton Vladimirovich Shutov
Vladimir State University named after Alexander Grigorievich and Nikolai Grigorievich Stoletov
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
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Review
For citations:
Zhukova A.A., Shutov A.V. Gelfond’s problem for expansions in linear recurrent bases. Chebyshevskii Sbornik. 2025;26(5):110-136. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-5-110-136
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