On some analogue of the Mahler-Eminyan problem for Ostrowsky expansions
https://doi.org/10.22405/2226-8383-2026-27-2-128-138
Abstract
A. O. Gelfond proved that for coprime 𝑏 − 1 and 𝑑, the sums of digits of 𝑏-ary expansions of natural numbers are uniformly distributed over arithmetic progressions with difference 𝑑. This result was later generalized to expansions over linear recurrent bases and to Ostrowsky expansions.
Eminyan solved the problem of the joint distribution modulo 2 of the sums of digits of binary expansions for pairs of consecutive natural numbers. Eminyan’s theorem can also be easily derived from one old result of Mahler. Later, a similar result was obtained for Zeckendorf
expansions over Fibonacci numbers and, further, for expansions over a wide class of linear recurrent bases.
In the present paper, we consider an analogue of the Mahler-Eminyan problem for Ostrowsky expansions associated with an arbitrary irrational 𝛼. Asymptotic formulas are obtained for the number of numbers not exceeding 𝑋, with a given distribution of the Ostrowsky expansions sums of digits of 𝑛 and 𝑛 + 1 modulo 2. The constants in the main term of the asymptotics are expressed as the sum of some infinite series, whose summands are expressed in terms of the
continued fraction expansion of 𝛼. These constants are explicitly calculated for 𝛼 = (√5−1)/2 .
The proof is based on the geometrization theorem previously obtained by the author and A. A. Zhukova, which describes sets of numbers with a fixed ending of the Ostrowsky expansion, as well as the theory of uniform distribution for the sequence {𝑛𝛼}.
About the Author
Anton Vladimirovich Shutov
Vladimir State University named after Alexander and Nikolay Stoletovs
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
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Review
For citations:
Shutov A.V. On some analogue of the Mahler-Eminyan problem for Ostrowsky expansions. Chebyshevskii Sbornik. 2026;27(2):128-138. (In Russ.) https://doi.org/10.22405/2226-8383-2026-27-2-128-138
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