Arithmetic Properties of Cantor Sets Involving Non-diagonal Forms

Abstract

We show conditions on k such that any number x in the interval $[0,{k\over 2}]$ can be represented in the form $x_{1}^{a_{1}}x_{2}^{a_{2}}+x_{3}^{a_{3}}x_{4}^{a_{4}}+\cdots+x_{k-1}^{a_{k-1}}x_{k}^{a_{k}}$, where the exponents a_2i−1 and a_2i are positive integers satisfying a_2i−1 + a_2i = s for $i=1,2,\ldots,{k\over 2}$, 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] AthreyaJS, ReznickB, TysonJT. Cantor set arithmetic. Amer. Math. Monthly, 2019, 126(1): 4-17

[2] CuiL, MaM. On arithmetic properties of Cantor sets. Sci. China Math., 2022, 65(10): 2035-2060

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

[4] HuaLKIntroduction to Number Theory, 1982, Berlin-New York. Springer-Verlag.

[5] PanCD, PanCBFundamentals of Analytic Number Theory, 1991, Beijing. Science and Technology Press. (in Chinese)

[6] RudinWReal and Complex Analysis, 1987Third EditionNew York. McGraw-Hill Book Co..

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

[8] WangZ, JiangK, LiW, ZhaoB. On the sum of squares of the middle-third Cantor set. J. Number Theory, 2021, 218: 209-222

[9] YeX, HuangW, ShaoSAn Introduction to Topological Dynamics, 2008, Beijing. Science Press. (in Chinese)

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Peking University