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

Abstract

LetW(k, 2) denote the, least numbers for which the system of equations$\sum\nolimits_{i = 1}^s {x_i^j = } \sum\nolimits_{i = 1}^s {y_i^j (1 \leqslant j \leqslant k)}$ has a solution with$\sum\nolimits_{i = 1}^s {x_i^{k + 1} \ne } \sum\nolimits_{i = 1}^s {y_i^{k + 1} }$. We show that for largek one hasW(k, 2)≦1/2k ²(logk+loglogk+O(1)), and moreover that whenK is large, one hasW(k, 2)≦1/2k(k+1)+1 for at least one valuek in the interval [K, K ^3/4+ε]. We show also that the leasts for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfiess≦k ²(logk+O(loglogk).