A quantitative Erd bs-Fuchs type result for multivariate linear forms

Let A be an infinite sequence of positive integers and let m = {(k1, m1), middot middot middot, (kl, ml)} be a set of integer tuples such that gcd(m1, m2, middot middot middot, ml) > 1. Denote by Rm(A, n) the number of different solutions of the equation k1(a1,1 + middot middot middot + a1,m1) + middot middot middot + kl(al,1 + middot middot middot + al,ml) <= n with ai, j E A. In 2013, Rue proved that for every epsilon > 0 the function Rm(A, n) = cn + O (n1/4-epsilon) cannot hold for any constant c > 0. Recently, we improved this bound to o(n1/4). In this paper, we obtain a stronger version about this result.(c) 2023 Elsevier B.V. All rights reserved.