Erdős's integer dilation approximation problem is resolved: sets with positive logarithmic density always contain distinct α, β with |nα−β| arbitrarily small.
Proving the Duffin-Schaeffer conjecture without GCD graphs
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We present a novel proof of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Our proof is heavily motivated by the ideas of Koukoulopoulos-Maynard's breakthrough first argument, but simplifies and strengthens several technical aspects. In particular, we avoid any direct handling of GCD graphs and their `quality'. We also consider the metric quantitative theory of Diophantine approximations, improving the $(\log \Psi(N))^{-C}$ error-term of Aistleitner-Borda and the first named author to $\exp(-(\log \Psi(N))^{\frac{1}{2} - \varepsilon})$.
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Erd\H{o}s's integer dilation approximation problem and GCD graphs
Erdős's integer dilation approximation problem is resolved: sets with positive logarithmic density always contain distinct α, β with |nα−β| arbitrarily small.