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arXiv · 1711.06422

Distribution of $αn + β$ modulo 1 over integers free from large and small primes

Kam Hung Yau
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Abstract

For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert αn + β\rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In addition, we also establish an asymptotic formula with an additional square-free condition on $n$. Moreover, if $α$ is quadratic irrational then the asymptotic formulas holds for all sufficiently large $x$. Our ingredients come from the Harman sieve which we adapt suitably to sieve for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for exponential sums.

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Kam Hung Yau. 2019-05-01. Distribution of $αn + β$ modulo 1 over integers free from large and small primes. https://arxiv.org/abs/1711.06422

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