The weak inhomogeneous Duffin–Schaeffer conjecture

1 · The essentials

The problem is about approximating a real number by multiplying it by positive integers. One fixes a shift and asks whether the result, after subtracting that shift, is close to an integer. The allowed error can vary freely with the multiplier, but each allowance must be finite and nonnegative. The manuscript claims that if a particular weighted sum of these allowances diverges, these close encounters happen infinitely often for almost every real number. Here “almost every” means that the exceptions form a set of measure zero, not that there are no exceptions. The shift can be any fixed real number, and the fractions associated with the approximations need not be in lowest terms. This result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

An imperfect image is a marked ruler on which you place successive multiples of a length, then move all the ruler’s marks by the same fixed amount. Each attempt has its own allowed margin around the marks: the manuscript gives a condition ensuring infinitely many successes for almost every starting length.

Why it matters

The significance of the claimed result is that it covers every fixed real shift without an additional arithmetic condition on that shift. It also allows irregular error allowances, without requiring them to increase or decrease. Its scope must nevertheless be distinguished from a statement requiring fractions in lowest terms: numerators here are unrestricted.

2 · Going further

Fix $γ \in R$ and a function $ψ : N \to [0, \infty)$ whose values are all finite. The manuscript claims the following implication: if $\sum_{q \geq 1} ϕ (q) ψ (q) / q = \infty$ , then for almost every $x$ , the inequality $∥ q x - γ ∥ < ψ (q)$ holds for infinitely many positive integers $q$ . Here $ϕ (q)$ is Euler’s totient function and $∥ y ∥$ is the distance from $y$ to the nearest integer.

The inequality means that there is an integer $p$ with $∣ q x - γ - p ∣ < ψ (q)$ , or equivalently $∣ x - (p + γ) / q ∣ < ψ (q) / q$ . There is no coprimality requirement on $p$ and $q$ . Although Euler’s totient appears in the divergence condition, the conclusion therefore does not restrict numerators to those coprime to the denominator. Neither monotonicity of $ψ$ nor a Diophantine restriction on $γ$ is required.

The source calls this statement the weak inhomogeneous Duffin–Schaeffer conjecture and claims its resolution in the manuscript. The supplied abstract does not describe the idea of the argument, so its method cannot be explained from this material. Only one manuscript is supplied; no relationship to companion manuscripts is specified.

Useful notions

Distance to the nearest integer
Divergence of a series with nonnegative terms
Coprime integers
Measure zero and “almost everywhere”

What was known before / what is new

Before

The source identifies the problem as the weak inhomogeneous Duffin–Schaeffer conjecture. It supplies no history, earlier results, or attribution for the formulation of the problem.

New

The manuscript claims that divergence of the weighted sum is sufficient for infinitely many approximations for almost every $x$ , for any fixed real shift. The statement accepts any finite-valued error function without monotonicity and does not require numerators to be coprime to denominators.

Limits and assumptions

The statement assumes a fixed shift $γ$ , a finite-valued nonnegative function $ψ$ , and divergence of $\sum_{q \geq 1} ϕ (q) ψ (q) / q$ . Its conclusion holds for almost every $x$ , not necessarily every $x$ ; it also does not assert that one full-measure set works simultaneously for all shifts. Numerators are unrestricted: a conclusion requiring reduced fractions is not supplied. The main result is not formalized in Lean and has not been formally verified. The source does not describe the argument or specify which questions remain open.

Mini glossary

Euler’s totient: The function $ϕ (q)$ counts the integers from $1$ to $q$ that are coprime to $q$ .
Distance to the nearest integer: The quantity $∥ y ∥$ is the smallest distance between the real number $y$ and an integer.
Almost everywhere: A property holds almost everywhere if the set on which it fails has measure zero.
Reduced fraction: A fraction is reduced when its numerator and denominator have no common divisor greater than $1$ .
Monotonicity: Monotonicity requires a function to vary in only one direction rather than alternating between increases and decreases.

Keywords : Inhomogeneous approximation · Duffin–Schaeffer · Almost everywhere

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that for every real shift γ and finite-valued $ψ : N \to [0, \infty)$ , divergence of $\sum_{q} ϕ (q) ψ (q) / q$ implies $∥ q x - γ ∥ < ψ (q)$ 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.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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The Weak Inhomogeneous Duffin–Schaeffer Conjecture

25/09/2026

To be verified

We prove the weak inhomogeneous Duffin–Schaeffer conjecture. For every fixed γ ∈ ℝ and finite-valued $ψ : N \to [0, \infty)$ , divergence of $\sum_{q \geq 1} ϕ (q) ψ (q) / q$ implies $∥ q x - γ ∥ < ψ (q)$ 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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