On \(q\)-ary periodic sequences. (English. Russian original) Zbl 1532.11100
J. Math. Sci., New York 276, No. 3, 384-386 (2023); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 179, 34-36 (2020).
Summary: We consider the problem of estimating the possible number of periods and the length of the periodic part of an irrational number depending on its measure of irrationality \(\beta \). We state that the expansion of the fractional part of an irrational number \(\alpha\) cannot start with a nonperiodic part of length \((1 - \delta)N\) and cannot terminate with a periodic part of length \( \delta N\), regardless of the numeral system.
MSC:
| 11J82 | Measures of irrationality and of transcendence |
References:
| [1] | A. A. Bukhshtab, Number Theory [in Russian], Lan’, Saint Petersburg (2008). · Zbl 0093.04601 |
| [2] | Chirskii, VG, Arithmetic properties of polyadic series with periodic coefficients, Dokl. Ross. Akad. Nauk, 459, 6, 677-679 (2014) · Zbl 1367.11066 |
| [3] | Chirskii, VG, On transformations of periodic sequences, Chebyshev. Sb., 17, 3, 180-185 (2016) · Zbl 1435.11100 |
| [4] | Chirskii, VG, Arithmetic properties of polyadic series with periodic coefficients, Izv. Ross. Akad. Nauk. Ser. Mat., 81, 2, 215-232 (2017) · Zbl 1376.11062 |
| [5] | Chirskii, VG, Periodic and nonperiodic finite sequences, Chebyshev. Sb., 18, 2, 275-278 (2017) · Zbl 1434.11149 |
| [6] | Chirskii, VG, Representation of positive integers by summands of a certain form, Proc. Steklov Inst. Math., 298, 1, 70-73 (2017) · Zbl 1383.11042 · doi:10.1134/S0081543817070045 |
| [7] | Chirskii, VG, Topical problems of the theory of transcendental numbers: Developments of approaches to their solutions in works of Yu. V. Nesterenko, Russ. J. Math. Phys., 24, 2, 153-171 (2017) · Zbl 1373.11002 · doi:10.1134/S1061920817020030 |
| [8] | Chirskii, VG; Nesterenko, AYu, On an approach to periodic sequences, Diskr. Mat., 27, 4, 150-157 (2015) · Zbl 1395.11107 |
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