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 sth powers. More generally, we study the existence of gaps between the values ⩽ N of diagonal forms of degree s in s variables with positive integer coefficients. We find: (1) gaps of size ≫ log N ( log log N ) 2 when s = 3 ; (2) gaps of size ≫ log log log N log log log log N 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 + 4 a ( c 3 x 3 ) 4 + 4 b ( c 4 x 4 ) 4 .