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 $f_i(x)$ of degree $k$ whose leading coefficients are relatively prime to $q$, there exists a solution $\underline{x}$ to the congruence $$ \sum_{i=1}^n f_i(x_i) \equiv c \pmod q $$ that lies in a cube of side length at least $\max\{q^{1/k},k\}$. 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 $k\geq 2$, $s\geq \lceil k(\log k+4.20032) \rceil$, and $\lambda_1,\dots ,\lambda_s,\omega\in\mathbb{R}$. Assume that the $\lambda_i$ 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 $$ |\lambda_1 x_1^{k}+\lambda_{2} x_2^{k}+\cdots+\lambda_{s} x_s^{k}+\omega|<\epsilon $$ has $\gg P^{s-k}$ integer solutions with $|x_i|\leq P$. Moreover the asymptotic formula for the number of smooth solutions is established assuming the same conditions hold.