Modular hyperbolas and Beatty sequences

Bounds for $\max\{m,\tilde{m}\}$ subject to $m,\tilde{m} \in \mathbb{Z}\cap[1,p)$, $p$ prime, $z$ indivisible by $p$, $m\tilde{m}\equiv z\bmod p$ and $m$ belonging to some fixed Beatty sequence $\{ \lfloor n\alpha+\beta \rfloor : n\in\mathbb{N} \}$ are obtained, assuming certain conditions on $\alpha$. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence $0,\, \operatorname{e}_p(y\overline{1}), \operatorname{e}_p(y\overline{2}), \ldots, \operatorname{e}_p(y(\overline{p-1}))$ on average are obtained, where $\operatorname{e}_p(t) = \exp(2\pi i t/p)$ and $m\overline{m} \equiv 1\bmod p$. The latter is accomplished by adapting a method due to Kloosterman.Comment: 13 pages, 1 figur

Higher-rank Bohr sets and multiplicative diophantine approximation

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.Comment: arXiv admin note: text overlap with arXiv:1703.0701

On the distribution of αp\alpha p modulo one in imaginary quadratic number fields with class number one

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.Comment: 36 page