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The weak inhomogeneous Duffin–Schaeffer conjecture

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.

Manuscripts in this family · 1

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

Sep 25, 2026·paper.pdf

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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