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

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