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 $\le N$ 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 $a(c_1{x}_1{)}^4+b(c_2{x}_2{)}^4+4a(c_3{x}_3{)}^4+4b(c_4{x}_4{)}^4$.

25 pages, preprint accepted upon revisions by the Journal of the London Mathematical Society