Result 022, Number theory

The weak inhomogeneous Duffin–Schaeffer conjecture

Proves that for every real shift γ and finite-valued ψ:N→[0,∞)\psi:\mathbb N\to[0,\infty), divergence of ∑qϕ(q)ψ(q)/q\sum_q\phi(q)\psi(q)/q implies ∥qx−γ∥<ψ(q)\|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.

Proof

The bigger picture

Why it matters

How often can multiplying a real number by an integer bring it close to a fixed shifted integer grid? The manuscript gives a broad condition guaranteeing infinitely many such encounters for almost every real number.

What changes?

The manuscript reports this for every fixed real shift gamma and every nonnegative, finite-valued error function psi: if the sum over positive integers q of phi(q) times psi(q) divided by q diverges, then, for almost every real x, q times x minus gamma is less than psi(q) away from an integer for infinitely many q. Here phi(q) counts integers from 1 to q coprime to q. "Almost every" means outside a set of measure zero.

What does that help mathematicians do?

The result lets researchers deduce infinitely many successful approximations even when the allowed errors fluctuate irregularly: psi need not be monotone, and gamma needs no special arithmetic properties. Under the divergence condition, failures cannot occupy a set of positive measure. The scope is deliberately weak: the integers used in the approximations need not be coprime to q, so this does not establish a version requiring reduced numerators.

Are there practical applications?

Its immediate value is foundational in number theory. It connects a sum weighted by coprimality counts to recurring approximation of almost every real number, despite allowing unrestricted numerators. This supplies a criterion for studying shifted rational approximation without imposing regularity on the error sizes or arithmetic restrictions on the shift.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Weak Inhomogeneous Duffin–Schaeffer Conjecture

September 25, 2026 87 pages

We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued ψ:N→[0,∞)\psi:\mathbb N\to[0,\infty), divergence of ∑q≥1ϕ(q)ψ(q)/q\sum_{q\ge1}\phi(q)\psi(q)/q implies ∥qx−γ∥<ψ(q)\|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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Cite (BibTeX) 
@misc{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026,
  author = {{OpenAI}},
  title = {{The Weak Inhomogeneous Duffin--Schaeffer Conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026/paper.pdf}{OAI:The-weak-inhomogeneous-Duffin-Schaeffer-conjecture-September-25-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.