Arithmetic Properties of Cantor Sets Involving Non-diagonal Forms

Abstract

We show conditions on k such that any number x in the interval

[0,\frac{k}{2}]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[0,{k\over 2}]$$\end{document}

can be represented in the form

x_1^{a_1}{x}_2^{a_2}+x_3^{a_3}{x}_4^{a_4}+\cdots +x_{k-1}^{a_{k-1}}{x}_k^{a_k}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x_{1}^{a_{1}}x_{2}^{a_{2}}+x_{3}^{a_{3}}x_{4}^{a_{4}}+\cdots+x_{k-1}^{a_{k-1}}x_{k}^{a_{k}}$$\end{document}

, where the exponents a_2i−1 and a_2i are positive integers satisfying a_2i−1 + a_2i = s for

i=1,2,\dots ,\frac{k}{2}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2,\ldots,{k\over 2}$$\end{document}

, and each x_i belongs to the generalized Cantor set. Moreover, we discuss different types of non-diagonal polynomials and clarify the optimal results in low-dimensional cases.

Keywords

Cantor set / Waring’s problem / non-diagonal forms / 28A80 / 11P05

References

[1] Athreya JS, Reznick B, Tyson JT. Cantor set arithmetic. Amer. Math. Monthly, 2019, 126(1): 4-17

[2] Cui L, Ma M. On arithmetic properties of Cantor sets. Sci. China Math., 2022, 65(10): 2035-2060

[3] Guo Y. Waring-Hilbert problem on Cantor sets. Expo. Math., 2021, 39(2): 165-181

[4] Hua LK. Introduction to Number Theory, 1982, Berlin-New York, Springer-Verlag

[5] Pan CD, Pan CB. Fundamentals of Analytic Number Theory, 1991, Beijing, Science and Technology Press(in Chinese)

[6] Rudin W. Real and Complex Analysis, 1987, Third Edition, New York McGraw-Hill Book Co.

[7] Steinhaus H. Mowa Własność Mnogości Cantora. Wector, 1917, 1: 1-3Selected Papers, Warsaw: PWN—Polish Scientific Publishers, 1985

[8] Wang Z, Jiang K, Li W, Zhao B. On the sum of squares of the middle-third Cantor set. J. Number Theory, 2021, 218: 209-222

[9] Ye X, Huang W, Shao S. An Introduction to Topological Dynamics, 2008, Beijing, Science Press(in Chinese)

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