Waring’s problem for finite intervals

Abstract

Let f ( n , k , s ) f(n,k,s) denote the cardinality of the smallest set A A of nonnegative k k -th powers such that every integer in [ 0 , n ] [0,n] is a sum of s s elements of A A , and let β ( k , s ) = lim su p n → ∞ log ⁡ f ( n , k , s ) / log ⁡ n \beta (k,s) = {\text {lim su}}{{\text {p}}_{n \to \infty }}\log f(n,k,s)/\log n . Clearly, β ( k , s ) ⩾ 1 / s \beta (k,s) \geqslant 1/s . In this paper it is proved that f ( n , k , s ) > c n 1 / ( s − g ( k ) + k ) f(n,k,s){\text { > }}c{n^{1/(s - g(k) + k)}} for all n ⩾ n 1 ( k , s ) n \geqslant {n_1}(k,s) , where g ( k ) g(k) is defined as in Waring’s problem, and β ( k , s ) ∼ 1 / s \beta (k,s) \sim 1/s as s → ∞ s \to \infty .