mathnasТруды Института математики НАН БеларусиProceedings of the Institute of Mathematics of the NAS of Belarus1812-5093Институт математики НАН Беларусиmathnas-63Research ArticleАЛГЕБРА И ТЕОРИЯ ЧИСЕЛALGEBRA AND NUMBER THEORYЗакон Бенфорда и приближение логарифмов натуральных чисел рациональнымиBenford’s law and approximation of logarithms of natural numbers by rational numbersБерникВ. И.BernikV. I.

Минск

Minsk

bernik.vasili@mail.ruКалошаН. И.KaloshaN. I.

Минск

Minsk

kalosha@im.bas-net.byВасильевД. B.VasilyevD. V.

Минск

Minsk

vasilyev@im.bas-net.byИнститут математики НАН БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus202530062025331714Copyright © Берник В.И., Калоша Н.И., Васильев Д.B., 20252025Берник В.И., Калоша Н.И., Васильев Д.B.Bernik V.I., Kalosha N.I., Vasilyev D.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathnas.ejournal.by/jour/article/view/63

В статье исследованы частотные свойства первых цифр в последовательности, образованной степенями целых чисел. Рассматривается ряд обобщений этой проблемы, а также обсуждается связь между распределением первых цифр и диофантовыми свойствами логарифмов. Предлагается ряд актуальных проблем в теории диофантовых приближений.

The paper is devoted to studying the frequencies at which first digits occur in series formed by powers of integer numbers. A number of generalizations of this problem are considered, and the relation between the distribution of first digits and Diophantine properties of logarithms is discussed. In conclusion of the article, several interesting problems in modern theory of Diophantine approximation are proposed.

диофантовы приближениязакон Бенфордараспределение первых цифрстепени целых чиселdiophantine approximationBenford's lawfirst digit distributionpowers of integersThis work was supported by the Institute of Mathematics of the National Academy of Sciences of Belarus within the framework of the state programme “Convergence–2020ReferencesNewcomb S. On the frequency of use of the different digits in natural numbers. American Journal of Mathematics, 1881, vol. 4, pp. 39-40.Newcomb S. On the frequency of use of the different digits in natural numbers. American Journal of Mathematics, 1881, vol. 4, pp. 39–40.Benford F. The law of anomalous numbers. Proceedings of the American Philosophical Society, 1938, vol. 78, pp. 551-572.Benford F. The law of anomalous numbers. Proceedings of the American Philosophical Society, 1938, vol. 78, pp. 551–572.Hurlimann W. Generalizing Benford’s Law Using Power Laws: Application to Integer ¨ Sequences. International Journal of Mathematics and Mathematical Sciences, 2009, art. 970284, 10 pp. https://doi.org/10.1155/2009/970284.Hurlimann W. Generalizing Benford’s Law Using Power Laws: Application to Integer ¨ Sequences. International Journal of Mathematics and Mathematical Sciences, 2009, art. 970284, 10 pp. https://doi.org/10.1155/2009/970284.Kuipers L., Niederreiter H. Uniform Distribution of Sequences. New York, Wiley, 1974, 390 p. (Pure and Applied Mathematics).Kuipers L., Niederreiter H. Uniform Distribution of Sequences. New York, Wiley, 1974, 390 p. (Pure and Applied Mathematics).Vinogradov I. M. Selected Papers. Berlin, Springer-Verlag, 1985, 401 p.Vinogradov I. M. Selected Papers. Berlin, Springer-Verlag, 1985, 401 p.Roth K. Rational approximations to algebraic numbers. Mathematica, 1955, vol. 2, pp. 1-20.Roth K. Rational approximations to algebraic numbers. Mathematica, 1955, vol. 2, pp. 1–20.Khinchine A. Zur metrischen Theorie der diophantischen Approximationen. Mathematische Annalen, 1924, vol. 92, pp. 115-125.Khinchine A. Zur metrischen Theorie der diophantischen Approximationen. Mathematische Annalen, 1924, vol. 92, pp. 115–125.Baker A., Wustholtz G. Linear forms in logarithms of algebraic numbers. ¨ Journal fur die reine ¨ und angewandte Mathematik, 1993, vol. 442, pp. 19-62. 14 V. I. Bernik, N. I. Kalosha, D. V. VasilyevBaker A., Wustholtz G. Linear forms in logarithms of algebraic numbers. ¨ Journal fur die reine ¨ und angewandte Mathematik, 1993, vol. 442, pp. 19–62. 14 V. I. Bernik, N. I. Kalosha, D. V. VasilyevHata M. Legendre type polynomials and irrationality measures. Journal fur die reine und ¨ angewandte Mathematik, 1990, vol. 407, pp. 99-125.Hata M. Legendre type polynomials and irrationality measures. Journal fur die reine und ¨ angewandte Mathematik, 1990, vol. 407, pp. 99–125.Rukhadze E. A. A problem on the distribution of first digits. Moscow University Mathematics Bulletin, 1987, vol. 42, no. 1, pp. 25-29 (in Russian).Rukhadze E. A. A problem on the distribution of first digits. Moscow University Mathematics Bulletin, 1987, vol. 42, no. 1, pp. 25–29 (in Russian).Baker A. Transcendental Number Theory. Cambridge, Cambridge University Press, 1990, 155 p.Baker A. Transcendental Number Theory. Cambridge, Cambridge University Press, 1990, 155 p.Bernik V., Kleinbock D., Margulis G. A. Khinchine-type theorems on manifolds: convergence case for standard and multiplicative versions. International Mathematics Research Notices, 2001, vol. 9, pp. 453-486.Bernik V., Kleinbock D., Margulis G. A. Khinchine-type theorems on manifolds: convergence case for standard and multiplicative versions. International Mathematics Research Notices, 2001, vol. 9, pp. 453–486.

The authors declare that there are no conflicts of interest present.