Erdős Problem #464

Let \(A=\{n_1<n_2<\cdots\}\subset \mathbb{N}\) be a lacunary sequence (so there exists some \(\epsilon>0\) with \(n_{k+1}\geq (1+\epsilon)n_k\) for all \(k\)). Must there exist an irrational \(\theta\) such that \[\{ \|\theta n_k\| : k\geq 1\}\] is not dense in \([0,1]\) (where \(\| x\|\) is the distance to the nearest integer)? Solved independently by de Mathan [dM80] and Pollington [Po79b], who showed that, given any such \(A\), there exists such a \(\theta\), with \[\inf_{k\geq 1}\| \theta n_k\| \gg \frac{\epsilon^4}{\log(1/\epsilon)}.\] This bound was improved by Katznelson [Ka01], Akhunzhanov and Moshchevitin [AkMo04], and Dubickas [Du06], before Peres and Schlag [PeSc10] improved it to \[\inf_{k\geq 1}\| \theta n_k\| \gg \frac{\epsilon}{\log(1/\epsilon)},\] and note that the best bound possible here would be \(\gg \epsilon\). This problem has consequences for [894](https://www.erdosproblems.com/894). The conclusion "\(\{\|\theta n_k\|\}\) is not dense in \([0,1]\)" is formalized as the sequence \((\theta n_k)\) not being dense modulo one; see the formalization notes above.