On Diophantine problems with mixed powers of primes
Volume 182 / 2018
Abstract
Let $k$ be an integer with $k\geq 3$ and $\varepsilon \gt 0$. Let $s(k)=[{(k+1)}/{2}]$ and $\sigma(k)=\min\bigl(2^{s(k)-1},\frac{1}{2}s(k)(s(k)+1)\bigr)$. Suppose that $\lambda_1,\lambda_2,\lambda_3$ are non-zero real numbers, not all negative, and $\lambda_1/\lambda_2$ is irrational and algebraic. Let $\mathcal{V}$ be a well-spaced sequence and $\delta \gt 0$. We prove that $E(\mathcal{V},X,\delta)\ll X^{1-{1}/{(8\sigma(k))}+2\delta+\varepsilon}$, where $E(\mathcal{V},X,\delta)$ denotes the number of $v\in \mathcal{V}$ with $1\leq v\leq X$ such that the inequality $|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^k-v| \lt v^{-\delta}$ has no solution in primes $p_1,p_2,p_3$. Furthermore, suppose that $\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5$ are non-zero real numbers, not all of the same sign, $\lambda_1/\lambda_2$ is irrational and $\varpi$ is a real number. We prove that there are infinitely many solutions in primes $p_j$ to the inequality $$|\lambda_1p_1^2+\lambda_2p_2^2+\lambda_3p_3^2+\lambda_4p_4^2+\lambda_5p_5^k+\varpi| \lt (\max p_j)^{-{1}/{(8\sigma(k))}+\varepsilon}.$$ This gives an improvement of an earlier result.
