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908 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
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tag “diophantine approximation” · 7 problems✕ clear
OPENPARTIALdiophantine approximation · number theory
Littlewood conjecture
Let \(\alpha,\beta \in \mathbb{R}\). Is it true that\[\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0\]? This is also known as the Littlewood conjecture.
For any function \(f:\mathbb{N}\to \mathbb{R}_{\geq 0}\) the property that, for almost all \(\alpha\) \[\left\lvert \alpha-\frac{p}{q}\right\rvert < \frac{f(q)}{q}\] has infinitely many solutions with \((p,q)=1\), is equivalent to…
PROVED LEANnumber theory · diophantine approximation
Let \(A=\{n_1<n_2<\cdots\}\) be an infinite sequence of positive integers, and let \(\phi_A(k)\) count the number of \(1\leq m\leq n_k\) such that the fraction \(\frac{m}{n_k}\) cannot be written as \(\frac{b}{n_j}\) for any integer \(b\)…
For any \(0<\alpha<1\), let \(f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})\). Does \(f(\alpha,n)\) have an asymptotic distribution function? In other words, is there a non-decreasing function \(g\) such…
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