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arxiv: 1907.04590 · v1 · pith:JENVRTHTnew · submitted 2019-07-10 · 🧮 math.NT

Decoupling theorems for the Duffin-Schaeffer problem

Christoph Aistleitner This is my paper

Pith reviewed 2026-05-24 23:46 UTC · model grok-4.3

classification 🧮 math.NT
keywords Duffin-Schaeffer conjecturemetric Diophantine approximationoverlap estimatesdecoupling theoremsKhintchine-type theoremsBorel-Cantelli lemmacoprime integers
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The pith

Upper bounds on overlaps show the Duffin-Schaeffer sets have weaker global dependence than supposed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Duffin-Schaeffer conjecture states that if the sum of the lengths of the intervals E_n diverges, then almost every x is in infinitely many E_n, where each E_n collects intervals around fractions a/n with gcd(a,n)=1. Controlling the overlaps between E_m and E_n for m ≠ n has been the main difficulty in proving the conjecture. This paper establishes new upper bounds on the measure of these intersections that hold uniformly for any non-negative ψ. The bounds demonstrate that the dependence among the sets is smaller than earlier estimates suggested, which in turn yields improved results for cases where the divergence is only slightly more than needed or grows slowly.

Core claim

We prove upper bounds for the measures of the pairwise overlaps E_m ∩ E_n which show that globally the degree of dependence in the set system (E_n)_{n ≥ 1} is significantly smaller than supposed. As applications, we obtain significantly improved 'extra divergence' and 'slow divergence' variants of the Duffin-Schaeffer conjecture.

What carries the argument

Uniform upper bounds on |E_m ∩ E_n| that depend only on the coprimality condition in the definition of the sets E_n.

If this is right

  • The extra divergence variant of the conjecture holds under weaker conditions than before.
  • The slow divergence variant of the conjecture holds under weaker conditions than before.
  • The global dependence among the sets E_n is significantly reduced compared to prior assumptions.
  • These bounds apply without any regularity assumptions on the function ψ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These decoupling results may simplify proofs of related statements in Diophantine approximation.
  • Similar techniques could apply to other metric problems involving dependent events in number theory.
  • Further work might extend the bounds to higher-order intersections for full Borel-Cantelli lemmas.

Load-bearing premise

The upper bounds on the overlap measures |E_m ∩ E_n| hold uniformly for all pairs m, n and all non-negative functions ψ, based solely on the coprimality definition.

What would settle it

A concrete counterexample consisting of specific values m, n and a function ψ where the measured overlap |E_m ∩ E_n| exceeds the derived upper bound would disprove the decoupling theorems.

read the original abstract

The Duffin-Schaeffer conjecture is a central open problem in metric number theory. Let $\psi~\mathbb{N} \mapsto \mathbb{R}$ be a non-negative function, and set $\mathcal{E}_n :=\bigcup \left( \frac{a - \psi(n)}{n},\frac{a+\psi(n)}{n} \right)$, where the union is taken over all $a \in \{1, \dots, n\}$ which are co-prime to $n$. Then the conjecture asserts that almost all $x \in [0,1]$ are contained in infinitely many sets $\mathcal{E}_n$, provided that the series of the measures of $\mathcal{E}_n$ is divergent. At the core of the conjecture is the problem of controlling the measure of the pairwise overlaps $\mathcal{E}_m \cap \mathcal{E}_n$, in dependence on $m, n, \psi(m)$ and $\psi(n)$. In the present paper we prove upper bounds for the measures of these overlaps, which show that globally the degree of dependence in the set system $(\mathcal{E}_n)_{n \geq 1}$ is significantly smaller than supposed. As applications, we obtain significantly improved "extra divergence" and "slow divergence" variants of the Duffin-Schaeffer conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves uniform upper bounds on the measures of the pairwise overlaps |ℰ_m ∩ ℰ_n| for the limsup sets ℰ_n arising in the Duffin-Schaeffer conjecture. These bounds hold for every pair m, n and every non-negative ψ, relying only on the coprimality definition of the intervals. The authors then apply the bounds to obtain improved 'extra divergence' and 'slow divergence' variants of the conjecture.

Significance. If the stated uniform bounds hold, the work materially reduces the apparent dependence among the sets ℰ_n and supplies new quantitative control that is independent of any regularity assumptions on ψ. This constitutes a concrete advance toward the Duffin-Schaeffer conjecture and supplies the first explicit decoupling estimates of this strength.

minor comments (2)
  1. [Abstract / Introduction] The abstract asserts that the degree of dependence is 'significantly smaller than supposed,' yet no explicit comparison with prior overlap estimates (e.g., those appearing in Harman or in the original Duffin-Schaeffer work) is supplied in the introduction.
  2. [Notation] Notation for the sets is introduced as ℰ_n in the abstract but appears as E_n in the reader's summary; a single consistent symbol should be used throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper, the recognition of the significance of the uniform decoupling bounds, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation consists of proving uniform upper bounds on |ℰ_m ∩ ℰ_n| that rely solely on the coprimality definition of the sets and hold for arbitrary non-negative ψ without additional regularity assumptions. These bounds are obtained as independent geometric estimates rather than by fitting parameters to the target divergence series or by re-expressing quantities already defined in terms of the conjecture. No self-citations are invoked as load-bearing steps for the core decoupling result, and the applications to extra-divergence and slow-divergence variants follow directly from the new bounds without circular reduction to the paper's inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies exclusively on standard axioms of Lebesgue measure and elementary number theory; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (2)
  • standard math Lebesgue measure on [0,1] is countably additive and translation-invariant
    Required to compute measures of the unions and intersections defining E_n and E_m ∩ E_n.
  • standard math The density of integers coprime to n equals φ(n)/n where φ is the Euler totient function
    Implicit in the definition of the sets E_n via the coprimality condition.

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Reference graph

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