Arbitrarily long gaps between the values of positive-definite cubic and biquadratic diagonal forms

Abstract

For $s=3,4$, we prove the existence of arbitrarily long sequences of consecutive integers none of which is a sum of $s$ nonnegative $s$-th powers. More generally, we study the existence of gaps between the values $\leq N$ of diagonal forms of degree $s$ in $s$ variables with positive integer coefficients. We find: (1) gaps of size $O\left(\frac{\sqrt {\log N}}{(\log \log N)^2}\right)$ when $s=3$; (2) gaps of size $O\left(\frac{\log\log\log N}{\log\log\log\log N}\right)$ if $s=4$ and the form, up to permutation of the variables, is not equal to $a (c_1x_1)^4+b (c_2 x_2)^4+4 a (c_3x_3)^4+4b(c_4x_4)^4$.