Arithmetic properties of Cantor sets involving non-diagonal forms

Abstract

We show conditions on $k$ such that any number $x$ in the interval $[0, k/2]$ 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 $i = 1, 2, \dots, k/2$, 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.