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Math-2808 · mathematics · Number theory

Weak inhomogeneous Duffin-Schaeffer

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix a real shift gamma and a finite-valued nonnegative function psi on positive integers. If $sum_q$ phi(q)psi(q)/q diverges, must almost every real x satisfy $||qx-\gamma ||<\psi (q)$ for infinitely many q, with no coprimality restriction on the approximating numerator?

Short English statement

Fix a real shift gamma and a finite-valued nonnegative function psi on positive integers. If $sum_q$ phi(q)psi(q)/q diverges, must almost every real x satisfy $||qx-\gamma ||<\psi (q)$ for infinitely many q, with no coprimality restriction on the approximating numerator?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2808-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2808

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 022; 2026-09-25).

Weak inhomogeneous Duffin-Schaeffer

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 022; 2026-09-25), the asserted result is that divergence of the totient-weighted approximation sum implies infinitely many inhomogeneous approximations for almost every real number, for every fixed shift and without reduced-numerator conditions.

Mathematical statement of the claimed result

Manuscript 1.
Fix any real $\gamma$ and any finite-valued function $\psi:\mathbb N\to[0,\infty)$. Writing $\phi$ for Euler's totient and $\|y\|=\min_{m\in\mathbb Z}|y-m|$, the manuscript claims \[ \sum_{q=1}^{\infty}\frac{\phi(q)}q\psi(q)=\infty \quad\Longrightarrow\quad \text{for Lebesgue-almost every }x\in\mathbb R, \quad\|qx-\gamma\|<\psi(q)\text{ for infinitely many }q. \] No monotonicity of $\psi$ is required. The approximating integer is unrestricted: this weak inhomogeneous statement imposes no coprimality condition on its numerator.

Outline of the argument

For manuscript 1 (family 022; 2026-09-25), close interval pairs are measured through determinant coordinates and weighted least common multiples. Exact rational coincidences are charged separately, while arithmetic extraction and a density purge give summable savings for the remaining shifted overlaps.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 022; 2026-09-25).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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