Some remarks on Vinogradov's mean value theorem and Tarry's problem

Abstract

Let W(k, 2) denote the, least number s for which the system of equations has a solution with . We show that for large k one has W(k, 2)≦1/2k 2 (log k +loglog k + O (1)), and moreover that when K is large, one has W(k, 2)≦1/2k(k+1)+1 for at least one value k in the interval [ K, K 3/4+ε ]. We show also that the least s for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfies s ≦ k 2 (log k + O (loglog k ).