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.
released 2026-09-25 | 1 theorem · 39 lemmas · 49 proofs · 41,844 words |
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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.