On one additive problem connected with expansions on linear recurrrent sequence
https://doi.org/10.22405/2226-8383-2023-24-3-228-241
Abstract
Let 𝑎1, . . . , 𝑎𝑑 be natural numbers satisfying condition 𝑎1 ≥ 𝑎2 ≥ . . . ≥ 𝑎𝑑−1 ≥ 𝑎𝑑 = 1.
Define sequence {𝑇𝑛} using the linear recurrent relation 𝑇𝑛 = 𝑎1𝑇𝑛−1 + 𝑎2𝑇𝑛−2 + . . . + 𝑎𝑑𝑇𝑛−𝑑 and initial conditions 𝑇0 = 1, 𝑇𝑛 = 1 + 𝑎1𝑇𝑛−1 + 𝑎2𝑇𝑛−2 + . . . + 𝑎𝑛𝑇0 for 𝑛 < 𝑑. Let N(𝑤) be a set of natural numbers for which the greedy expansion on the linear recurrent sequence {𝑇𝑛} ends with some word 𝑤. Here 𝑤 is chosen in such a way that so that the set N(𝑤) is non-empty.
We study the problem about the number 𝑟𝑘(𝑁) of representations of a natural number 𝑁 in as the sum of 𝑘 terms from N(𝑤).
Using the previously obtained description of the sets N(𝑤) in terms of shifts of tori of dimension 𝑑 − 1, an asymptotic formula for the number of representations 𝑟𝑘(𝑁) is obtained, and also found upper bounds for the number of representations.
Conditions on 𝑘 that ensure the existence of considered representations for all sufficiently large natural numbers 𝑁 are found. In particular, such representations exist if 
, where 𝑚 is the length of the fixed end 𝑤 of the greedy expansion.
In addition, an asymptotic formula is obtained for the average number of representations.
In conclusion, several unsolved problems are formulated.
About the Author
Anton Vladimirovich Shutov
Khabarovsk Division of the Institute for Applied Mathematics,
Far Eastern Branch, Russian Academy of Sciences
Russian Federation
Russian Federation
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Review
For citations:
Shutov A.V. On one additive problem connected with expansions on linear recurrrent sequence. Chebyshevskii Sbornik. 2023;24(3):228-241. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-3-228-241
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