Applications of the Hardy-Littlewood method to polynomial congruences and Diophantine inequalities

Abstract

This manuscript revolves around two peer-reviewed results. First, we prove that for any positive integers k, q, n with n>N(k), integer c, and polynomials fi(x) of degree k whose leading coefficients are relatively prime to q, there exists a solution x― to the congruence ∑i=1nfi(xi)≡c(modq)that lies in a cube of side length at least max{q1/k,k}. Moreover, the result is best possible up to the determination of N(k). The latter half of the manuscript is centred around Diophantine inequalities. Let k≥2, s≥⌈k(log⁡k+4.20032)⌉, and λ1,…,λs,ω∈R. Assume that the λi are non-zero, not all in rational ratio, and not all of the same sign in the case that k is even. Then, for any $\epsilon > 0 $, the inequality |λ1x1k+λ2x2k+⋯+λsxsk+ω|<ϵ$$has$≫Ps−k$integersolutionswith$|xi|≤P$.Moreovertheasymptoticformulaforthenumberofsmoothsolutionsisestablishedassumingthesameconditionshold.