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 a2i−1 and a2i are positive integers satisfying a2i−1 + a2i = s for \(i=1,2,\ldots,{k\over 2}\) , and each xi belongs to the generalized Cantor set. Moreover, we discuss different types of non-diagonal polynomials and clarify the optimal results in low-dimensional cases.
