Arithmetic properties of Cantor sets involving non-diagonal forms

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