On one additive problem connected with expansions on linear recurrrent sequence

A. V. Shutov

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 228-241.

Résumé

Let $a_1,\ldots,a_d$ be natural numbers satisfying condition $a_1\geq a_2\geq\ldots\geq a_{d-1}\geq a_d=1.$ Define sequence $\{T_n\}$ using the linear recurrent relation $T_n=a_1T_{n-1}+a_2T_{n-2}+\ldots+a_dT_{n-d}$ and initial conditions $T_0=1,$ $T_n=1+a_1T_{n-1}+a_2T_{n-2}+\ldots+a_nT_0$ for $n$. Let $\mathbb{N}(w)$ be a set of natural numbers for which the greedy expansion on the linear recurrent sequence $\{T_n\}$ ends with some word $w$. Here $w$ is chosen in such a way that so that the set $\mathbb{N}(w)$ is non-empty. We study the problem about the number $r_k(N)$ of representations of a natural number $N$ in as the sum of $k$ terms from $\mathbb{N}(w)$. Using the previously obtained description of the sets $\mathbb{N}(w)$ in terms of shifts of tori of dimension $d-1$, an asymptotic formula for the number of representations $r_k(N)$ 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 $k\geq 1+(a_1+1)^{m-d+1}\frac{(a_1+1)^d-1}{a_1}$, 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.

Détail

Keywords: linear recurrent sequences, greedy expansions, fixed last digits, linear additive problem.

Bibliographie

[1] Drmota M., Gajdosik J., “The Parity of the Sum-of-Digits-Function of Generalized Zeckendorf Representations”, Fibonacci Quarterly, 36:1 (1998), 3–19 | DOI | MR | Zbl

[2] Lamberger M., Thuswaldner J. W., “Distribution properties of digital expansions arising from linear recurrences”, Mathematicf Slovaca, 53:1 (2003), 1–20 | MR | Zbl

[3] Shutov A., “On sum of digits of the Zeckendorf representations of two consecutive numbers”, Fibonacci Quarterly, 58:3 (2020), 203–207 | MR | Zbl

[4] Shutov A. V., “On one sum associated with Fibonacci numeration system”, Far Eastern Mathematical Journal, 20:2 (2020), 271–275 (in Russian) | DOI | MR | Zbl

[5] Drmota M., Gajdosik J., “The distribution of the sum-of-digits function”, Journal de Theorie des Nombres de Bordeaux, 10:1 (1998), 17–32 | DOI | MR | Zbl

[6] Zhukova A.A., Shutov A.V., “On Gelfond-type problem for generalized Zeckendorf representations”, Chebyshevskii Sbornik, 22:2 (2021), 104–120 (in Russian) | DOI | DOI | MR | Zbl

[7] Zhuravlev V. G., “Fibonacci-even numbers: Binary additive problem, distribution over progressions, and spectrum”, St. Petersburg Mathematical Journal, 20:3 (2009), 339–360 | DOI | MR | Zbl

[8] Davletjarova E. P., Zhukova A. A., Shutov A. V., “Geometrization of the Fibonacci numeration system, with applications to number theory”, St. Petersburg Mathematical Journal, 25:6 (2014), 893–907 | DOI | MR

[9] Davletjarova E. P., Zhukova A. A., Shutov A. V., “Geometrizacija obobshhennyh sistem schislenija Fibonacci i ee prilozhenija k teorii chisel”, Chebyshevskii sbornik, 17:2 (2016), 88–112 (in Russian) | DOI | MR | Zbl

[10] Zhukova A. A., Shutov A. V., “Geometrization of numeration systems”, Chebyshevskii Sbornik, 18:4 (2017), 222–245 | DOI | MR | Zbl

[11] Parry W., “On the $\beta$-expansions of real numbers”, Acta Math. Acad. Sci. Hungar., 11:3 (1960), 401–416 | DOI | MR | Zbl

[12] Shutov A.V., “Generalized Rauzy tilings and linear recurrence sequences”, Chebyshevskii Sborni, 22:2 (2021), 313–333 (in Russian) | DOI | DOI | MR | Zbl

[13] Shutov A. V., “On one additive problem with fractional parts”, Nauchn. ved. BelGU. Ser. math. phys., 5(148):30 (2013), 111–120

[14] Zhukova A.A., Shutov A., “Additive Problem with $k$ Numbers of a Special Form”, Journal of Mathematical Sciences, 2022, 163–174 | DOI | DOI

[15] Niederreiter H., Wile M., “Diskrepanz und Distanz von Maßlen bezüglich konvezer und Jordanscher Mengen”, Math. Z., 144:1 (1975), 125–134 | DOI | MR | Zbl

[16] Götz M., “Discrepancy and error in integration”, Monatsh. Math., 136 (2002), 99–121 | DOI | MR | Zbl

[17] Macbeath A.M., “On measure of sum sets II: the sum theorem for the torus”, Mathematical Proceedings of the Cambridge Philosophical Society, 49:1 (1953), 40–43 | DOI | MR | Zbl

[18] Frougny C., Solomyak B., “Finite beta-expansions”, Ergodic Theory and Dynamical Systems, 12:4 (1992), 713–723 | DOI | MR | Zbl

[19] Gricenko S. A., Mot'kina N. N., “Zadacha Hua-Lokena s prostymi chislami special'nogo vida”, DAN respubliki Tadzhikistan, 52:7 (2009), 497–500 (in Russian)

[20] Gricenko S. A., Mot'kina, N. N., “On Chudakov's theorem involving primers of a special type”, Chebyshevskii sbornik, 12:4 (2011), 75–84 (in Russian) | MR | Zbl

[21] Gricenko S.A., Mot'kina N. N., “Ob odnom variante ternarnoj problemy Gol'dbaha”, DAN respubliki Tadzhikistan, 52:6 (2009), 413–417 (in Russian)

[22] Gricenko S.A., Mot'kina N. N., “Waring's promblem involving natural numbers of a special type”, Chebyshevskii sbornik, 15:3 (2014), 31–47 (in Russian) | MR | Zbl