Some remarks on Vinogradov's mean value theorem and Tarry's problem
Let \(k\) be a positive integer and \(W(k)\) denote the least number \(s\) for which the system of equations \[ \sum^s_{i=1} x_i^j= \sum_{i=1}^s y_i^j, \qquad j=1,\dots, k, \] has a solution with \(\sum_{i=1}^s x_i^{k+1}\neq \sum_{i=1}^s y_i^{k+1}\). With the help of his recent results in [\textit{T. D. Wooley}, Mathematika 39, 379--399 (1992; Zbl 0769.11036); J. Am. Math. Soc. 7, 221--245 (1994; Zbl 0786.11053) and in Number theory with an emphasis on the Markoff spectrum, Lect. Notes Pure Appl. Math. 147, 317--321 (1993; Zbl 0791.11014)], the author proves that for large \(k\) \[ W(k)\leq \tfrac12 k^2(\log k+\log\log k+O(1)), \] and that for each \(\varepsilon>0\) there is a real \(K(\varepsilon)\) with the property that for each \(K\geq K(\varepsilon)\) there exists a \(k\) in the interval \([K, K^{4/3+\varepsilon}]\) with \[ W(k)\leq \tfrac12 k(k+1)+1. \] He also shows 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 \(s\leq k^2(\log k+O(\log\log k))\).
scientific article; zbMATH DE number 1051737
scientific article