The weak inhomogeneous Duffin–Schaeffer conjecture

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Proves that for every real shift γ and finite-valued \(\psi:\mathbb N\to[0,\infty)\), divergence of \(\sum_q\phi(q)\psi(q)/q\) implies \(\|qx-\gamma\|\lt \psi(q)\) for infinitely many q, for almost every x. Here ϕ is Euler’s totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.

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Theorem 1.1 (Weak inhomogeneous Duffin–Schaeffer)

In The Weak Inhomogeneous Duffin–Schaeffer Conjecture, 25 September 2026

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Let \gamma\in\mathbb R be fixed, and let \psi:\mathbb N\to[0,\infty) be finite-valued. If

\sum_{q=1}^{\infty}\frac{\phi(q)}q\,\psi(q)=\infty,

then, for Lebesgue-almost every x\in\mathbb R, the inequality

\|qx-\gamma\|<\psi(q)

holds for infinitely many positive integers q.

Corollary 1.2 (Hausdorff-measure divergence)

In The Weak Inhomogeneous Duffin–Schaeffer Conjecture, 25 September 2026

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Fix \gamma\in\mathbb R and a finite-valued \psi:\mathbb N\to[0,\infty). Let f:(0,\infty)\to(0,\infty) be continuous and nondecreasing, with f(r)\to0 as r\downarrow0, and suppose that f(r)/r is monotone. Set f(0)=0 and

W_\gamma(\psi)=\{x\in\mathbb R: \|qx-\gamma\|<\psi(q)\text{ for infinitely many }q\}.

\sum_{q=1}^{\infty}\phi(q)f\!\left(\frac{\psi(q)}q\right)=\infty,

then, for every bounded interval I,

\mathcal H^f(I\cap W_\gamma(\psi))=\mathcal H^f(I),

where \mathcal H^f denotes Hausdorff f-measure.

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The Weak Inhomogeneous Duffin–Schaeffer Conjecture

25 September 2026, current edition.

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We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued \(\psi:\mathbb N\to[0,\infty)\), divergence of \(\sum_{q\ge1}\phi(q)\psi(q)/q\) implies \(\|qx-\gamma\|\lt \psi(q)\) for infinitely many q, for almost every x. Here ϕ is Euler’s totient and \(\|\cdot\|\) is distance to the nearest integer. Numerators need not be reduced, and the shift satisfies no Diophantine restriction.

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19 references in The Weak Inhomogeneous Duffin–Schaeffer Conjecture

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