REVIEW 1 major objections 4 minor 14 references
General Duffin--Schaeffer-type counterexamples in diophantine approximation
T0 review · 1 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash
Pith's one-line read For every slowly diverging approximation function f, a restriction of the denominators to a sparse set S turns the well-approximable numbers into a null set, giving Khintchine-type counterexamples that also work inhomogeneously.
desk verdict A strong, novel construction in metric Diophantine approximation; main theorem is new, proof is basically right, but a factor-of-two inconsistency and an unproved monotonicity claim need fixing. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is the set S' = X·Y, where Y(I) consists of products of one prime from each pair (p_{2i-1}, p_{2i}) and X is an arithmetic progression modulo a large multiple P of all primes involved. Lemma 2.2 converts the measure of the union of intervals A_{xy} into a sum weighted by g_I(y)f(xy); Lemma 2.3 guarantees that one can choose I so that very few y have g_I(y) large, thanks to a lower bound on the number of prime pairs in dyadic intervals. For the inhomogeneous part, Lemma 3.2 reduces A_q^γ(ε) ⊆ A_{bq}(2bε) whenever bγ is close to an integer, allowing the homogeneous blocks to be reused.
What would settle it
Compute the quantity #Y(I,δ)/#Y times (1/Σ_{y∈Y} y_min/y) for the set I constructed in Lemma 2.3 with δ = 0.01 and K = 50; the proof predicts this is below 0.01. If the ratio is actually above δ, then Lemma 2.3 is false and the block construction of Proposition 2.1 collapses, removing the proof of Theorem 1.2.
Extended reading notes
Core claim
The central discovery is a block construction: for any non-increasing f satisfying f(q) → 0 and ∑ f(q) = ∞, one can find finite sets S_k of denominators whose approximation intervals A_q nearly do not overlap, so λ(⋃_{q∈S_k} A_q) < ε while ∑_{q∈S_k} f(q) is bounded away from zero. Unioning these blocks yields S with divergent sum and λ(W(f1_S)) = 0. The construction is flexible enough to force monotonicity on the support and to carry over to inhomogeneous shifts γ that are very close to rationals with denominators b_k; a covering lemma embeds each inhomogeneous interval into a homogeneous one. A concrete special case gives, for f_k(q) = 1/(q log q ··· $log^{{(k)}}$ q), a sufficient wildly-Liouville condition for γ under which a denominator set produces a null well-approximable set.
Load-bearing premise
The construction relies on the prime number theorem estimate that every interval [e^j, $e^{{j+1}}$) contains at least e^j/(50j) pairs of consecutive primes p_{2i-1}, p_{2i} with gap at most 40j for all large j; if this estimate fails for infinitely many j, the blocks Y(I) can no longer be forced to have the overlap pattern needed to make the union measure small.
Editorial extensions
If this is right
- If Theorem 1.2 holds, the Khintchine zero–full dichotomy is false for non-monotone ψ even under the bound ψ(q) ≤ f(q) for every divergent f with f(q) → 0, in particular for f(q) = 1/q.
- Theorem 1.3 provides counterexamples that are monotone on the support and whose support has upper density 1 at infinitely many scales; this is the strongest possible in view of the Duffin–Schaeffer theorem, which forbids positive lower density.
- In the inhomogeneous setting, for each f there is an uncountable Γ containing ℚ such that every γ ∈ Γ is a zero-measure shift for some S, so Yu's tame-Liouville condition cannot be weakened to plain irrationality.
- For the explicit functions f_k, Theorem 1.6 gives a concrete wildly-Liouville criterion in terms of iterated exponentials, and the authors note that the gap to Yu's tame class is probably not closable by optimizing the proof.
- The paper leaves open the quantitative refinement that ψ(q) ≪ 1/q might still imply full measure for tame Liouville shifts.
Reading between the lines
- The same block construction might transfer to higher-dimensional or systems-of-linear-forms settings, since the overlap control is measure-theoretic once Y is chosen; a testable extension would be to verify whether an analogous prime-pair estimate holds in number fields or for polynomial denominators.
- The near-disjointness of intervals is achieved probabilistically, which suggests a general recipe: choose a denominator set with large pairwise gcds to control overlaps; one could test this on random multiplicative sets outside the prime-pair construction.
- The wildly-Liouville notion attached to f in Theorem 3.4 is defined through the function H from Proposition 2.1; making H explicit for other f, beyond the f_k family, would yield quantitative inhomogeneous criteria similar to Theorem 1.6.
- The negative answer to the Erdős–Vaaler question suggests that the correct condition for the zero–one law in the non-monotone case is not a size bound but a structural property of the support S; one could investigate whether a natural 'gcd-dense' condition on S is sufficient.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper constructs Duffin–Schaeffer-type counterexamples for monotone approximation functions. Given any non-increasing f: N -> [0, infinity) with f(q) -> 0 and divergent series, the authors find S ⊆ N with ∑_{q∈S} f(q) = infinity but λ(W(f 1_S)) = 0 (Theorem 1.2), and refine this to a function ψ ≤ f that is non-increasing on a support of full upper density (Theorem 1.3). In the inhomogeneous setting, they produce uncountably many shifts γ, including Q, for which analogous zero-measure counterexamples exist (Theorem 1.5), and a quantitative version for iterated-logarithm functions with an explicit wildly-Liouville condition (Theorem 1.6). The proofs use a block construction with prime-pair denominators to force destructive overlaps among approximation intervals, plus Borel–Cantelli arguments.
Significance. If correct, the homogeneous theorem resolves the question whether Khintchine's theorem can fail for ψ bounded by any non-increasing f with divergent series, extending the classical Duffin–Schaeffer counterexample to essentially optimal generality. The full-density and inhomogeneous variants answer questions raised by Erdős, Vaaler, and Yu. The constructions are novel, especially the monotonicity-on-support feature, and the paper contains explicit effective bounds for the iterated-logarithm family. The proof is largely self-contained and uses only standard tools (prime number theorem, Hoeffding's inequality, Borel–Cantelli); the deep Koukoulopoulos–Maynard theorem appears only in a remark. However, one essential monotonicity assertion in the block construction is not proved.
major comments (1)
- [§2.1.3] The assertion 'Since f ≥ 0 is non-increasing, so too is F(y)/y' (appearing just after the definition of F(y)) is not proved and is load-bearing. It is used to pass from the sum over y ∈ Y to the bound ∑_{y∈Y(I,δ)} g_I(y) F(y)/y ≤ #Y(I,δ) F(y_min)/y_min + δ ∑_{y∈Y} F(y)/y, and then to (2.10). If this monotonicity fails, the estimate λ(⋃_{q∈S} A_q) < ε collapses. The claim is true (a discrete layer-cake argument gives it), but a rigorous proof must be supplied in the paper before the central construction is complete.
minor comments (4)
- [Proposition 2.1] The statement in (2.1) records ∑_{q∈S} λ(A_q) ∈ [1,2], but the proof's Step (3) chooses ∑ f(q) ∈ [1,2]; since λ(A_q) = 2f(q), the statement and proof are off by a factor of two. The divergence conclusion is unaffected, but the statement should be corrected (e.g., to [2,4]) or the proof should choose ∑ f(q) ∈ [1/2,1].
- [Theorem 3.4] The existence of H with max S ≤ H(1/ε) in Proposition 2.1 (for M=1) is asserted without justification. Since the proof of Proposition 2.1 is constructive, such an H can be extracted, but the paper should state this explicitly or supply a bound.
- [Lemma 2.3] The display (2.4) is ambiguous in the typeset version; it should be written as a ratio: #Y(I,δ) / (∑_{y∈Y} y_min/y) < δ. The surrounding text confirms this interpretation, but the notation should be clarified.
- [§2.1.3, Step (1)] The asymptotic notation o_Y(1) is used without a formal definition. It is understandable in context, but a brief definition would improve readability.
Circularity Check
No circularity: the main constructions are self-contained and do not reduce to their inputs.
full rationale
The derivation is not circular. Theorem 1.2 is proved by assembling finite blocks S_k whose existence is established in Proposition 2.1 from the given f via a product construction X·Y: Y is chosen from Lemma 2.3 using only the prime number theorem and prime-gap estimates, and X is chosen in Step (3) to make sum_{x in X, y in Y} f(xy) lie in [1,2]. The small-union bound in Lemma 2.2 uses only monotonicity of f(q)/q and the group structure of Y; no equation defining S uses W(f 1_S) or lambda(W(f 1_S)). The external estimates (prime number theorem, Hoeffding, Mertens) are parameter-free inputs, not fitted values. The inhomogeneous results reduce to the homogeneous block construction via Lemma 3.2 and the measure-preserving map x -> bx, which is a reduction to an independent statement, not circularity. Two non-circular issues deserve flagging. First, the claim 'Since f >= 0 is non-increasing, so too is F(y)/y' (Section 2.1.3) is load-bearing for (2.10)-(2.11) but is asserted without proof; an omitted proof is a correctness gap, not a self-reference. Second, Proposition 2.1 states sum_{q in S} lambda(A_q) in [1,2] while Step (3) selects sum f(xy) in [1,2]; since lambda(A_q) = 2 f(q), the statement and construction differ by a factor 2. This is a constant inconsistency, but it does not feed a conclusion back into an input. Self-citations are motivational and external; Koukoulopoulos-Maynard appears only in a remark and is not load-bearing for the main theorems.
Assumptions & free parameters
assumptions (6)
- standard math Prime Number Theorem
- standard math Mertens' first theorem
- standard math Hoeffding's inequality
- standard math Borel-Cantelli lemma
- standard math Measure-preserving property of multiplication by b modulo 1
- standard math Koukoulopoulos-Maynard theorem (Duffin-Schaeffer conjecture)
Cite this review
Pith. "Pith review of General Duffin--Schaeffer-type counterexamples in diophantine approximation." pith.science (2026). https://pith.science/paper/BR434QB4
@misc{pith2026250416565,
author = {Pith},
title = {Pith review of: General Duffin--Schaeffer-type counterexamples in diophantine approximation},
year = {2026},
howpublished = {\url{https://pith.science/paper/BR434QB4}},
note = {Machine review of arXiv:2504.16565}
}
read the original abstract
Duffin and Schaeffer provided a famous counterexample to show that Khintchine's theorem fails without monotonicity assumption. Given any monotonically decreasing approximation function with divergent series, we construct Duffin--Schaeffer-type counterexamples by restricting the denominator. We also extend these constructions to the inhomogeneous setting. Our results resolve some natural questions arising from the works of Erd\H{o}s, Vaaler, and Yu.
Reference graph
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Reviewed August 16, 2026 · model on record in the stance chip above.
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