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arXiv · 1706.00343

A Diophantine approximation problem with two primes and one $k$-th power of a prime

Alessandro Gambini·Alessandro Languasco·Alessandro Zaccagnini
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Abstract

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$-th power of a prime which was only proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ by combining Harman's technique on the minor arc with a suitable estimate for the $L^4$-norm of the relevant exponential sum over primes $S_k$. In the common range we also give a stronger bound for the approximation.

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Alessandro Gambini, Alessandro Languasco, Alessandro Zaccagnini. 2017-05-31. A Diophantine approximation problem with two primes and one $k$-th power of a prime. https://doi.org/10.1016/j.jnt.2018.01.002

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