Beresnevich–Velani's Hausdorff-measure Duffin–Schaeffer conjecture

Let ψ:NR\psi:\mathbb{N}\to\mathbb{R} be non-negative, let W(ψ)W(\psi) be the associated limsup set of reduced rational approximation intervals, and let f:[0,)[0,)f:[0,\infty)\to[0,\infty) be a dimension function. Write Hf\mathcal H^f for the corresponding Hausdorff ff-measure on R/Z\mathbb{R}/\mathbb{Z}. Beresnevich–Velani's conjecture. If r1f(r)r^{-1}f(r) is monotonic and

n=1f(ψ(n)n)φ(n)=,\sum_{n=1}^{\infty}f\left(\frac{\psi(n)}{n}\right)\varphi(n)=\infty,

then

Hf(W(ψ))=Hf(R/Z).\mathcal H^f(W(\psi))=\mathcal H^f(\mathbb{R}/\mathbb{Z}).

The paper states that this conjecture is equivalent to the classical Duffin–Schaeffer conjecture via the Mass Transference Principle, and it was open in the source.

Sources & referencesView supporting material

Primary source

Liangpan Li, “The Duffin-Schaeffer type conjectures in various local fields”, arXiv:1401.0035 (2013).

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