A critical majorant for the Khinchin-Ostrowski property

Bartosz Malman

REVIEW 3 major objections 3 minor 11 references

A critical majorant for the Khinchin-Ostrowski property

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves the exponential growth rate exp(c/(1-|z|)) is the exact threshold where Khinchin-Ostrowski uniqueness sets must contain intervals, and below which interval-free sets can still be uniqueness sets.

desk verdict Removes Khrushchev's integrability assumption and identifies the exact critical majorant; core proof sound, Section 4 typos are cosmetic. read the letter →

arxiv 2506.06911 v1 pith:KX4DAOVY submitted 2025-06-07 math.CV math.CA

Bartosz Malman This is my paper

classification math.CVmath.CA MSC 30C8530D5530H2030E10

keywords Khinchin-OstrowskipropertyuniquenesssetsBeurling-CarlesonconditionharmonicmeasureJoukowski-Privalovdomainexponentialmajorantone-sidedspectraldecaylocallogarithmicintegrability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to identify the sharp boundary for the Khinchin-Ostrowski property, the phenomenon whereby slowly growing polynomials that vanish on a suitably fat subset of the unit circle are forced to vanish throughout the disk. It proves that the exponential majorants $\exp(c/(1-|z|))$ are exactly that boundary: for every majorant strictly below them, of the form $\lambda_h(1/(1-|z|))=\exp(h(1-|z|)/(1-|z|))$ with mild regularity on $h$, there exists a closed, interval-free set $E$ of positive Lebesgue measure for which the property holds; in fact any $h$-Beurling-Carleson set of positive measure with no intervals works. At or above the exponential rate, the property can hold only if $E$ contains an interval. This sharpening removes a logarithmic-integrability hypothesis that had blocked earlier proofs for slowly decaying $h$, and it yields a precise optimality statement for a one-sided Fourier decay condition in a local Jensen-type inequality.

What carries the argument

The mechanism is the Joukowski-Privalov domain $D_E$, built by deleting from the unit disk the regions enclosed by the hyperbolic geodesics joining the endpoints of each interval complementary to $E$. Harmonic measure on this domain is estimated through the Joukowski map $\phi_L(z)=\frac{L}{1-L^2}(\frac{L}{z}+\frac{z}{L})$, which sends the model domain $\Omega_L=\mathbb{H}\setminus\{z:|z|\le L\}$ onto the upper half-plane and turns geodesic subarcs into intervals whose harmonic measure at $0$ is computed explicitly (Lemma 2.1). The load-bearing estimate, Proposition 3.1, is that $\int_{\partial D_E\cap\mathbb{D}} \frac{h(1-|z|^2)}{1-|z|^2}\,d\omega(z) \le C\sum_{\ell\in C(E)} h(|\ell|)$, which is exactly the integrability against harmonic measure needed to run Khrushchev's method without the logarithmic-integrability assumption (1.4).

What would settle it

To falsify the main claim it suffices to find one function $h$ satisfying $h(0)=0$ and the monotonicity conditions (Reg) for which every closed interval-free set $E$ of positive Lebesgue measure has $\sum_{\ell\in C(E)} h(|\ell|)=\infty$; then no $h$-Beurling-Carleson set of the required kind exists and Theorem 1.2 has no content. A concrete candidate to check is $h(t)=1/\log(1/t)$, for which the finiteness of the sum over interval-free sets is nontrivial and can be investigated directly.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.2, is that for every continuous increasing $h$ with $h(0)=0$ and $h(x)/x$ decreasing (no further size or integrability condition), and for every closed set $E\subset\mathbb{T}$ of positive Lebesgue measure containing no intervals that satisfies the $h$-Beurling-Carleson condition $\sum_{\ell\in C(E)} h(|\ell|)<\infty$, the pair $(\lambda_h,E)$ has the Khinchin-Ostrowski property. The paper also proves Theorem 1.3: given any positive sequence $c_n\to 0$, there is a nonzero Borel function $f$ whose carrier $\{z\in\mathbb{T}: f(z)\neq 0\}$ contains no intervals and whose one-sided Fourier coefficients obey $|\hat f(n)|=O(\exp(-c_n\sqrt n))$. Together these results show that the exponential majorants are critical in the sense that they are the smallest growth restrictions for which the Khinchin-Ostrowski property is governed solely by interval containment, and that the exponent $c\sqrt n$ in the one-sided spectral condition used to detect local logarithmic integrability cannot be improved to $c_n\sqrt n$ with $c_n\to 0$ without losing the conclusion.

Load-bearing premise

The load-bearing premise is that for every admissible $h$ with $h(0)=0$, there exists a closed set $E\subset\mathbb{T}$ of positive Lebesgue measure, containing no intervals, with $\sum_{\ell\in C(E)} h(|\ell|)<\infty$; the paper states that such a set is readily constructed but supplies no proof, and without one Theorem 1.2 would be vacuous for that $h$.

Editorial extensions

If this is right

For every majorant below the exponential family, interval-free closed sets of positive measure can be uniqueness sets, so the Khinchin-Ostrowski property is not determined by interval containment alone in this range.
For every majorant at least as large as $\exp(c/(1-|z|))$, a uniqueness set must contain an interval, closing the dichotomy.
Khrushchev's uniqueness theorem extends to all $\lambda_h$ with $h$ satisfying (Reg), with the auxiliary condition $\int_0^1 h(t)/t\,dt<\infty$ removed.
In the weighted setting, the exponential majorants are likewise the critical point for whether the Khinchin-Ostrowski property is determined by integrability of $\log w$ over intervals alone.
The one-sided spectral decay bound $|\hat f(n)|=O(\exp(-c\sqrt n))$ for local logarithmic integrability is sharp, since arbitrary $c_n\to 0$ in place of a fixed $c>0$ permits interval-free counterexamples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test, left implicit in the paper, is whether the regularity condition (Reg) is necessary; if it is dropped, the dichotomy between interval-free uniqueness and interval-dependent uniqueness may split into intermediate regimes.
The explicit harmonic-measure estimates for geodesic corners could be transferred to other approximation problems, such as Carleson measures for weighted Bergman spaces or polynomial approximation with prescribed growth near the boundary.
Theorem 1.3 quantifies the cost of weakening the spectral decay: the slower the sequence $c_n$, the thinner the support one can arrange, suggesting a quantitative trade-off between decay and carrier mass that the paper does not extract.
One could push the construction to ask for the minimal $h$ needed for a prescribed $c_n$; the piecewise-constant $h$ in Lemma 4.3 is a candidate, and the answer would locate the sharp threshold at a finer scale than the exponential majorant.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bartosz Malman has written a short note that resolves a real gap. Khrushchev's 1978 uniqueness theorem needed the logarithmic integrability condition (1.4); this paper removes it and proves Theorem 1.2 for the full range of majorants (1.1), with the exponential majorants exp(c/(1-|z|)) as the sharp threshold. That is a genuine advance, not a repackaging. The main technical content is Proposition 3.1, the harmonic measure estimate in the Joukowski-Privalov domain. The proof is careful and the dyadic decomposition argument is clean; I checked the steps and they hold. The application to one-sided spectral decay (Theorem 1.3) sharpens the earlier counterexample from exponent a<1/2 to the exact exp(-c_n sqrt(n)) with c_n→0, which is the right endpoint.

The soft spots are concentrated in Section 4 and are essentially typographical. Equation (4.1) defines h on intervals (c_{n+1}√(n+1), c_n√n], but under condition (iii) the left endpoint is larger than the right; the intended formula is c_{n+1}/√(n+1). Lemma 4.1's proof sets ec_1=c_1 even when later c_m exceed c_1; a max over m fixes that. The dual pairing in the proof of Theorem 1.3 needs the conjugate kernel, otherwise \hat f(n) is off by a reflection. These are all repairable and none touch the main theorem.

Two smaller caveats. The paper asserts after (1.2) that one may readily construct an h-Beurling-Carleson interval-free set of positive measure; a referee will want that spelled out, but it is an easy construction. And Theorem 1.2 is said to follow from Proposition 3.1 by 'Khrushchev's technique'; the paper does not give the Egorov-type argument. For a specialist note that is fair, but a few more sentences would make it self-contained.

The citations to Malman's own earlier work are legitimate: they are inputs, not conclusions. No circularity.

Bottom line: this is a solid subfield paper. The mathematics is correct as far as I can see, and the contribution is real. Send it to a referee; it should be accepted after the Section 4 typos are fixed.

Referee Report

3 major / 3 minor

Summary. The paper studies the Khinchin-Ostrowski (KO) property for majorants of the form λ_h(1/(1-|z|)) = exp(h(1-|z|)/(1-|z|)), where h is increasing, h(0)=0, h(x)/x→∞, and h(x)/x is decreasing. Its main theorem, Theorem 1.2, asserts that for every such h there is a closed, interval-free set E of positive Lebesgue measure, satisfying the h-Beurling-Carleson condition Σ_{ℓ∈C(E)} h(|ℓ|)<∞, for which the pair (λ_h,E) has the KO property. The proof is based on a new Joukowski-Privalov domain D_E obtained by removing hyperbolic-geodesic regions from the disk; Proposition 3.1 gives an explicit, self-contained estimate showing that the majorant is integrable against the harmonic measure of D_E without assuming the logarithmic integrability condition (1.4) needed in Khrushchev's theorem. This identifies the family exp(c/(1-|z|)) as a critical threshold below which interval-free uniqueness sets exist and above which an interval is necessary. The paper also states Theorem 1.3, an application to one-sided spectral decay: if |f̂(n)| = O(exp(-c_n√n)) with c_n→0, then f can vanish locally almost nowhere while its carrier is interval-free, sharpening the known exp(-c√n) condition.

Significance. If the Section 4 proof is repaired, the paper makes a valuable contribution: it removes an extraneous integrability hypothesis from a classical uniqueness theorem and gives a clean critical-majorant characterization. The core estimate, Proposition 3.1, is elementary and explicitly checkable, and it is proved without fitting any parameter to the conclusion. The use of Joukowski-type conformal mappings is natural and likely to be of independent use. The paper also gives a concrete Fourier-analytic application showing the sharpness of exp(-c√n) spectral decay, which is an interesting external benchmark. The main claims are falsifiable in the sense that the h-Beurling-Carleson condition and the KO property are explicitly defined, and the construction of E is explicit once the missing existence proof is supplied.

major comments (3)

[Section 4, Eq. (4.1)] The definition h(x)=c_n^2 for x∈(c_{n+1}√(n+1), c_n√n] is inconsistent with the surrounding proof. Under the stated condition (iii) of Lemma 4.1 the sequence c_n√n is nondecreasing, so the intervals (c_{n+1}√(n+1), c_n√n] are empty or reversed. Moreover, the derivative argument in Lemma 4.3 requires the right endpoint of the interval to be c_m/√m, not c_m√m, for the inequality x < c_m/√n to hold for all m≥n. The intended formula appears to be h(x)=c_n^2 for x∈(c_{n+1}/√(n+1), c_n/√n]. As written, the proof of Theorem 1.3 does not go through; this needs a careful correction.
[Section 4, Lemma 4.1] The recursive definition ec_{n+1}=max(max_{m≥n+1} c_m, √(n/(n+1)) ec_n) ensures condition (iii) as a lower bound: ec_{n+1} ≥ √(n/(n+1)) ec_n, equivalently ec_n√n ≤ ec_{n+1}√(n+1). Therefore the statement immediately after Lemma 4.1 that the sequence c_n√n is 'clearly decreasing to 0' is false and is also inconsistent with the later use of (iii) in Lemma 4.3, which requires c_n√n to be nondecreasing. The author should clarify whether the monotonicity statement concerns c_n/√n instead, and ensure that the endpoint monotonicity needed for the intervals in (4.1) is proved or assumed explicitly.
[After Eq. (1.2)] The assertion that for every h satisfying (Reg) and h(x)/x→∞ one may readily construct a closed, interval-free set E of positive Lebesgue measure with Σ_{ℓ∈C(E)} h(|ℓ|)<∞ is used as the existence input for Theorem 1.2, but no proof or reference is given. If such a set did not exist for some allowed h, the theorem would be vacuous. The reader's suggested construction (choose lengths with h(|ℓ_n|)=2^{-n}, place the intervals densely, and use h(x)/x→∞ to make the total length finite) is short and should be included, at least as a remark.

minor comments (3)

[Section 3.2, proof of Proposition 3.1] There is a typographical slip in the display 'L t_{t+1}/2' where 't_{n+1}' is intended; also the inequality 'sup_{w∈B_{L,n}} ...' is written with an unbalanced parenthesis in the text just after (3.4), which should be corrected.
[Section 3.1] The remark that adding a finite number of points to E does not change membership in any h-Beurling-Carleson class is not justified for general h, since splitting one complementary interval ℓ into two intervals can change Σ h(|ℓ|). For the usual monotone h the total h-mass can only increase, so it would be safer to state the needed normalization explicitly or restrict to subadditive-like h.
[Section 1.3] The passage from Proposition 3.1 to Theorem 1.2 is described as a direct application of Khrushchev's technique, but the manuscript does not spell out the Egorov/limsup argument that converts L^1(E)-convergence of p_n to 0 into the conclusion that p_n(z)→0 for all z∈D. Since this is the mechanism linking the harmonic-measure estimate to the KO property, a few sentences or a lemma would improve readability and verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the critical-majorant theorem is proved against external benchmarks; self-citations are inputs, not fitted conclusions.

full rationale

The derivation chain is self-contained in the sense that matters. Theorem 1.2 is a constructive/conditional theorem: for any h satisfying (Reg), Proposition 3.1 estimates harmonic measure in a Joukowski-Privalov domain and shows that the h-Beurling-Carleson condition (1.2) forces the required integrability (1.7); the Khinchin-Ostrowski property then follows by Khrushchev's external argument. No parameter in Proposition 3.1 is fitted to the existence of E or to the Khinchin-Ostrowski conclusion. The complementary statement that majorants at least as large as exp(c/(1-|z|)) force an interval is drawn from Khrushchev's theorem and the author's prior weighted results as independent inputs, not from Theorem 1.2 itself. The unproved assertion after (1.2) that an h-Beurling-Carleson set exists for every admissible h is a real completeness gap, but it is not circular: it is easily supplied by choosing complementary intervals with h(|\ell|)=2^{-n}, and Theorem 1.2 is stated conditionally on such a set existing. Section 4 contains repairable technical defects, notably the initialization in Lemma 4.1 and the interval definition in (4.1), which appear to use c_n\sqrt{n} where c_n/\sqrt{n} is needed; these are correctness risks in the proof of Theorem 1.3, not reductions of the conclusion to its own inputs. Overall, no load-bearing step equates the target result with an assumed or fitted quantity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation uses only standard harmonic measure and functional analytic facts, together with the stated regularity hypotheses on h and an unproved existence assertion for h-BC sets. No new particles, forces, or fitted constants are introduced.

assumptions (6)

standard math Standard properties of harmonic measure: monotonicity under domain inclusion, mutual absolute continuity with arclength on rectifiable boundaries.
Invoked in Section 1.3 and Lemma 3.2 via [Ran95, Cor. 4.3.9] and the Riesz brothers theorem.
standard math Mergelyan's theorem: polynomials in z are dense in C(E) for compact E with connected complement.
Section 4.2 uses this to conclude f=0 from vanishing moments against all polynomials.
domain assumption For any h satisfying (Reg), h(0)=0, there exists a closed positive-measure set E containing no intervals with sum h(|ℓ|) finite.
Stated without proof after (1.2); needed for Theorem 1.2 to be non-vacuous.
domain assumption The constant C in the growth condition (i) of Definition 1.1 is uniform in n.
The proof and Khrushchev's technique require one C for the whole sequence; the text is ambiguous.
standard math Khrushchev's technique: integrability (1.7) of the majorant against harmonic measure turns L1(E) convergence into pointwise convergence near 0.
Used after Proposition 3.1; details cited to [Khr78, Section 3].
standard math Completeness of monomials z^n in P_G^2.
Used in Section 4.2 to infer g=0 from vanishing moments.

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Cite this review

Pith. "Pith review of A critical majorant for the Khinchin-Ostrowski property." pith.science (2026). https://pith.science/paper/KX4DAOVY

@misc{pith2026250606911,
  author       = {Pith},
  title        = {Pith review of: A critical majorant for the Khinchin-Ostrowski property},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KX4DAOVY}},
  note         = {Machine review of arXiv:2506.06911}
}

abstract

In this short note we prove an optimal version of a classical result. Given a majorant determining a growth restriction on functions in the unit disk $\mathbb{D}$, we say that a set $E$ on the unit circle $\mathbb{T}$ is a uniqueness set, or has the Khinchin-Ostrowski property, with respect to the majorant, if any sequence of analytic polynomials satisfying the growth restriction which converges in an appropriate sense to $0$ on $E$, in fact is forced to converge to $0$ in $\mathbb{D}$ also. Theorems proved by Kegejan and Khrushchev state that if $E$ has positive Lebesgue measure and satisfies a generalized Beurling-Carleson condition, then for an appropriate majorant the Khinchin-Ostrowski property is satisfied. A technical point in Khrushchev's proof is the estimation of the harmonic measure in a Privalov-type domain which requires logarithmic integrability of the majorant. This forbids the application of his result to certain types of generalized Beurling-Carleson conditions. Here, we dispose of the integrability assumption on the majorant. To do so, we use a Joukowski-Privalov domain which is obtained by removing from the unit disk the areas enclosed by hyperbolic geodesics between the endpoints of intervals complementary to $E$. For this type of domain the method of Khrushchev applies, but the harmonic measure may be estimated more accurately by simple explicit formulas for conformal mappings. As a consequence, we find the critical majorant at which Beurling-Carleson type conditions stop determining the Khinchin-Ostrowski property of a set, and above which the containment of intervals is the only relevant characteristic. We discuss also weighted versions of the Khinchin-Ostrowski property, and apply our result to establish the remarkable precision of a one-sided spectral decay condition which detects the local logarithmic integrability of a function.

Figures

Figures reproduced from arXiv: 2506.06911 by the authors.

**Figure 1.** Domain used in Khrushchev’s proof. Observe, however, that the sides of the squares Sℓ make a right angle with T, and elementary formulas for conformal mappings onto domains with corners show that the harmonic measure near a corner should be significantly smaller than the arclength measure. In fact, we would expect ω(I) ≲|ℓ| |I| 2 whenever I is a sufficiently short segment of ∂Sℓ touching T (see [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 2.** The action of the mapping ϕL between ΩL and H. 2.2. Harmonic measure of subarcs. Consider the arc AL,t = {Leiθ : 0 ≤ θ ≤ t}, t < π/2. This is a subarc of AL of length |AL,t| = Lt lying in the semicircular part of the boundary of ΩL, with right end-point at L ∈ R ∩ ∂ΩL. See the red markings in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** An example of a domain DE, with the set E marked in blue. The well-known upper half-plane harmonic measure formula ω(H, B, i) = 1 π Z B 1 t 2 + 1 dt which holds for Borel subsets B ⊂ R, readily implies the desired estimate: ω(ΩL, AL,t, i) = 1 π Z 2L 1−L2 2L cos(t) 1−L2 1 t 2 + 1 dt ≤ 1 π Z 2L 1−L2 2L cos(t) 1−L2 1 dt = (1 − cos(t)) 2L π(1 − L2 ) . □ 3. Integrability properties of harmonic measure in Joukowski-Prival… view at source ↗

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Works this paper leans on

11 extracted references · 10 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Beurling

A. Beurling. On quasianalyticity and general distributions. (unpublished work) , 1961

work page 1961

[2] [2]

Borichev and A

A. Borichev and A. Volberg. Uniqueness theorems for almost analytic functions. Leningrad Mathematical Journal , 1:157--190, 1990

work page 1990

[3] [3]

Havin and B

V. Havin and B. J \"o ricke. The uncertainty principle in harmonic analysis , volume 28. Springer Science & Business Media, 2012

work page 2012

[4] [4]

E. M. Kegejan. Simultaneous approximation in a disc. Izv. Akad. Nauk Armjan. SSR Ser. Mat , (1):317--330, 1966

work page 1966

[5] [5]

S. V. Khrushchev. The problem of simultaneous approximation and of removal of the singularities of C auchy type integrals. Trudy Matematicheskogo Instituta imeni VA Steklova , 130:124--195, 1978

work page 1978

[6] [6]

P. K. Kythe. Handbook of conformal mappings and applications . Chapman and Hall/CRC, 2019

work page 2019

[7] [7]

Levinson

N. Levinson. Gap and density theorems , volume 26. American Mathematical Soc., 1940

work page 1940

[8] [8]

B. Malman. Revisiting mean-square approximation by polynomials in the unit disk. Accepted for publication in Journal d'Analyse Mathématique, arXiv preprint arXiv:2304.01400 , 2023

work page Pith review arXiv 2023