In their previous work devoted to the distribution of rational points near Brownian motion, the authors conjectured the existence of an \emph{oscillation principle} governing the asymptotic behavior of the number of rational points with bounded denomi\-nators near the graph of a monofractal curve. In this note, a weake...
The classical three-gap theorem says that a finite Kronecker sequence on the circle has at most three gap lengths. We extend this phenomenon to nearest-neighbour distances on quotients $(V\times U)/\Lambda$, where $V$ is a finite-dimensional real normed space, $U$ is an arbitrary ultrametric abelian group and $\Lambda$...
Given a real-valued function $f$, let $\mathcal{N}_f(\delta, Q)$ be the number of rational points with denominators at most $Q\ge 1$ in the $(\delta/Q)$-tubular neighbourhood of the graph of the function $f$. A heuristic predicts that the number of such points grows like the area of the neighbourhood provided that $\de...
We prove that if the exceptional set $E_p$ for the $p$-adic Littlewood conjecture is non-empty, then its logarithmic Hausdorff dimension is at least one. More precisely, whenever $E_p$ is non-empty, it has positive Hausdorff measure with respect to the gauge function $ h(r)=\frac{1}{\log(1/r)}. $ In particular, every n...
D. Badziahin, Volodymyr Pavlenkov, E. Zorin· 0 citations
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