Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets

Natanael Alpay; Paata Ivanisvili

arxiv: 2502.16045 · v2 · submitted 2025-02-22 · 🧮 math.CA · math.CO

Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets

Natanael Alpay , Paata Ivanisvili This is my paper

Pith reviewed 2026-05-23 02:59 UTC · model grok-4.3

classification 🧮 math.CA math.CO

keywords dyadic square functionsindicator functionslower boundsBrownian motionTakagi functionBurkholder-Davis-Gundymartingalesexit times

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The pith

The L1 norm of the dyadic square function S2 of any indicator 1_A is bounded below by the expected square root of Brownian motion exit time starting at |A|.

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

This paper derives sharp lower bounds on the L1 norms of dyadic square functions S1 and S2 when applied to indicator functions of measurable sets in the unit interval. For S2, the bound involves the expectation under Brownian motion of the square root of its first exit time from the interval (0,1) when started at the measure of A. This yields an asymptotic lower bound of order |A| times the logarithm of 1 over |A|, providing a logarithmic improvement to the classical Burkholder-Davis-Gundy lower bound. For S1 the corresponding sharp bound is given by the value of the Takagi function at |A|. The results apply to arbitrary measurable sets with no further assumptions.

Core claim

For every measurable A subset of [0,1), the inequality ||S2(1_A)||_1 >= E_{|A|}[sqrt(tau)] holds, where tau is the first exit time from (0,1) of Brownian motion started at |A|, and this quantity is comparable to |A|* log2(1/|A|*) with |A|* = min{|A|,1-|A|}. A similar sharp inequality holds for S1 with the Takagi function T(|A|).

What carries the argument

The dyadic square functions S1 and S2 defined with respect to the standard dyadic filtration on [0,1), lower-bounded via Brownian exit time expectations and the Takagi function.

If this is right

The classical Burkholder-Davis-Gundy lower bound of order |A|* is improved by a logarithmic factor.
The bounds are sharp and achieved asymptotically for suitable choices of A.
The same logarithmic improvement holds for both the quadratic and linear versions of the square function.
The results require no regularity assumptions on the set A beyond measurability.

Where Pith is reading between the lines

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

These bounds may extend to square functions on other filtrations or in higher dimensions.
The explicit Takagi function bound for S1 could be used to derive new estimates for certain lacunary series.
The Brownian motion comparison invites testing whether analogous exit-time expressions appear for other diffusion processes.

Load-bearing premise

The dyadic square functions are defined using the standard dyadic intervals on [0,1) and the lower bounds apply to all measurable sets without additional structure.

What would settle it

A measurable set A where ||S2(1_A)||_1 is smaller than a constant multiple of E[sqrt(tau)] by an arbitrarily large factor would disprove the lower bound.

Figures

Figures reproduced from arXiv: 2502.16045 by Natanael Alpay, Paata Ivanisvili.

read the original abstract

We study lower bounds for dyadic square functions of indicator functions. In the case of the dyadic square function $S_{2}$ we obtain a sharp lower bound: for every measurable $A \subset {[0,1)}$, we have \[ \|S_{2}(\mathbbm{1}_{A})\|_{1}\ge \mathbb{E}_{|A|}\big[\sqrt{\tau}\big]\asymp |A|^{*}\log_2\frac{1}{|A|^{*}}, \] where $\tau$ is the first exit time from $(0,1)$ of a standard Brownian motion started at $|A|$, and $|A|^{*}:=\min\{|A|,1-|A|\}$. This estimate gives logarithmic improvement over the classical Burkholder--Davis--Gundy lower bound $|A|^{*}$. In addition, we show a sharp inequality \[ \|S_{1}(\mathbbm{1}_{A})\|_{1} \ge T(|A|)\asymp |A|^{*}\log_{2}\frac{1}{|A|^{*}}, \] where $T(x)=\sum_{k=0}^{\infty}\frac{\operatorname{dist}(2^{k}x,\mathbb{Z})}{2^{k}}$ is the Takagi function.

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.

Desk Editor's Note private letter to a colleague

The paper pins down explicit sharp constants for the L1 norms of dyadic S2 and S1 on indicators using Brownian exit time and the Takagi function.

read the letter

The main thing here is that the authors give an explicit sharp lower bound for ||S2(1_A)||_1 equal to the expected square root of the Brownian exit time from |A|, which is asymptotically |A| log(1/|A|), and a parallel bound for S1 via the Takagi function. This is a logarithmic improvement on the classical Burkholder-Davis-Gundy lower bound of order |A|* and the expressions are new for these operators on indicators. The paper does well by making the constants concrete and tying them to standard objects like Brownian motion and the Takagi function, while stating the result for arbitrary measurable A with no extra regularity assumptions. The stress-test note confirms the claim is internally consistent and does not rely on hidden structure. The soft spots are limited: the abstract states the result cleanly but the full proofs are needed to check the lower bound and sharpness arguments, and the work stays inside the dyadic setting on [0,1), which is the right scope but keeps it specialized. No circularity or fitting issues appear. This is for people working on martingale and dyadic inequalities who care about sharp constants. A reader in that niche gets value from the explicit forms and the log sharpening. It deserves a serious referee because the claim is new, the approach uses reproducible techniques, and the statement holds up on its own terms. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes sharp lower bounds for the L^1 norms of dyadic square functions S_2 and S_1 applied to indicator functions 1_A of arbitrary measurable sets A ⊂ [0,1). For S_2 it proves ||S_2(1_A)||_1 ≥ E_{|A|}[√τ] ≍ |A|^* log_2(1/|A|^*), where τ is the first exit time from (0,1) of Brownian motion started at |A| and |A|^* = min{|A|,1-|A|}; this improves the classical BDG lower bound by a logarithmic factor. A parallel sharp bound ||S_1(1_A)||_1 ≥ T(|A|) ≍ |A|^* log_2(1/|A|^*) is given, with T the Takagi function.

Significance. If the claimed inequalities hold, the work supplies explicit, sharp constants realized by Brownian exit-time expectations and the Takagi function, furnishing a logarithmic improvement over the Burkholder–Davis–Gundy inequality that is attained asymptotically by indicator functions. The explicit probabilistic and functional representations constitute a concrete advance in the study of dyadic martingale inequalities.

minor comments (2)

The precise definitions of the dyadic square functions S_1 and S_2 (including the underlying filtration and normalization) should be recalled in §1 or §2 for readers outside the immediate subfield.
The notation |A|^* is introduced in the abstract but its relation to the starting point of the Brownian motion is not restated in the statement of the main theorem; a single sentence of clarification would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report accurately summarizes the main results on sharp lower bounds for the dyadic square functions S_2 and S_1 applied to indicators.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes explicit lower bounds on dyadic square function norms for arbitrary measurable indicator functions using standard martingale definitions and comparisons to Brownian exit times and the Takagi function. These are derived inequalities, not reductions to fitted parameters, self-definitions, or self-citation chains. The asymptotic equivalences follow from known properties of the referenced distributions and functions, with no load-bearing steps that collapse to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

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discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel echoes

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ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Bα,β(x) … satisfying … Bα((x+y)/2) ≤ ½[(Bβ(x)+|x-y|/2^β)^{α/β} + …] … B(0)=B(1)=0

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Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Burkholder, B

D.L. Burkholder, B. Davis, R.F. Gundy,Integral inequalities for convex functions of operators on martingales , Proc. 6th Berkeley Symp. Math. Statistics and Probability, 2 (1972) pp. 223–240

work page 1972
[2]

Burkholder, R.F

D.L. Burkholder, R.F. Gundy,Extrapolation and interpolation for convex functions of operators on martingales Acta Math. , 124 (1970) pp. 249–304

work page 1970
[3]

Durcik, P

Durcik, P. , Ivanisvili, P. , and Joris R..Sharp isoperimetric inequalities on the Ham- ming cube near the critical exponent. arXiv preprint arXiv:2407.12674 (2024)

work page arXiv 2024
[4]

Beltran, P

Beltran, D., Ivanisvili, P., and Madrid, J. (2023).On sharp isoperimetric inequalities on the hypercube. arXiv [Math.CA]. Retrieved from http://arxiv.org/abs/2303.06738

work page arXiv 2023
[5]

Wang, Sharp Square-Function Inequalities for Conditionally Symmetric Martin- gales, TAMS Vol

G. Wang, Sharp Square-Function Inequalities for Conditionally Symmetric Martin- gales, TAMS Vol. 328, No. 1 (Nov., 1991), pp. 393–419 (27 pages)

work page 1991
[6]

Wang, Sharp inequalities for the conditional square function of a martingale , The Annals of Probability 1991, Vol

G. Wang, Sharp inequalities for the conditional square function of a martingale , The Annals of Probability 1991, Vol. 19, No. 4, 1679-1688

work page 1991
[7]

On the L p norms of stochastic integrals and other martingales , Duke Mathematical Jurnal Vol

Davis, Burgess. On the L p norms of stochastic integrals and other martingales , Duke Mathematical Jurnal Vol. 43, No. 4 (1976): 697-704

work page 1976
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Hytönen, J

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Berlin: Springer, 2016

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The Annals of Probability 25.1 (1997): 206-214

Bobkov, Sergey G.An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. The Annals of Probability 25.1 (1997): 206-214

work page 1997
[11]

(2012).Sharp martingale and semimartingale inequalities

Osękowski, A. (2012).Sharp martingale and semimartingale inequalities . (Vol. 72) Springer Science & Business Media

work page 2012
[12]

Hart, S. (1976). A note on the edges of the n-cube. Discrete Mathematics, 14(2), 157–163

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[13]

Stein, E. M. (1970). Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63). Princeton University Press. (NA) Department of Mathematics, Universiy of California Ir vine, Ir vine, CA 92697, USA Email address: nalpay@uci.edu (PI) Department of Mathematics, Universiy of California Ir vine, Ir vine, CA 92697, USA Email address: pivanisv@uci.edu

work page 1970

[1] [1]

Burkholder, B

D.L. Burkholder, B. Davis, R.F. Gundy,Integral inequalities for convex functions of operators on martingales , Proc. 6th Berkeley Symp. Math. Statistics and Probability, 2 (1972) pp. 223–240

work page 1972

[2] [2]

Burkholder, R.F

D.L. Burkholder, R.F. Gundy,Extrapolation and interpolation for convex functions of operators on martingales Acta Math. , 124 (1970) pp. 249–304

work page 1970

[3] [3]

Durcik, P

Durcik, P. , Ivanisvili, P. , and Joris R..Sharp isoperimetric inequalities on the Ham- ming cube near the critical exponent. arXiv preprint arXiv:2407.12674 (2024)

work page arXiv 2024

[4] [4]

Beltran, P

Beltran, D., Ivanisvili, P., and Madrid, J. (2023).On sharp isoperimetric inequalities on the hypercube. arXiv [Math.CA]. Retrieved from http://arxiv.org/abs/2303.06738

work page arXiv 2023

[5] [5]

Wang, Sharp Square-Function Inequalities for Conditionally Symmetric Martin- gales, TAMS Vol

G. Wang, Sharp Square-Function Inequalities for Conditionally Symmetric Martin- gales, TAMS Vol. 328, No. 1 (Nov., 1991), pp. 393–419 (27 pages)

work page 1991

[6] [6]

Wang, Sharp inequalities for the conditional square function of a martingale , The Annals of Probability 1991, Vol

G. Wang, Sharp inequalities for the conditional square function of a martingale , The Annals of Probability 1991, Vol. 19, No. 4, 1679-1688

work page 1991

[7] [7]

On the L p norms of stochastic integrals and other martingales , Duke Mathematical Jurnal Vol

Davis, Burgess. On the L p norms of stochastic integrals and other martingales , Duke Mathematical Jurnal Vol. 43, No. 4 (1976): 697-704

work page 1976

[8] [8]

Hytönen, J

T. Hytönen, J. V. Neerven, M. Veraar, and L. Weis.Analysis in Banach spaces. Vol

work page

[9] [9]

Berlin: Springer, 2016

work page 2016

[10] [10]

The Annals of Probability 25.1 (1997): 206-214

Bobkov, Sergey G.An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. The Annals of Probability 25.1 (1997): 206-214

work page 1997

[11] [11]

(2012).Sharp martingale and semimartingale inequalities

Osękowski, A. (2012).Sharp martingale and semimartingale inequalities . (Vol. 72) Springer Science & Business Media

work page 2012

[12] [12]

Hart, S. (1976). A note on the edges of the n-cube. Discrete Mathematics, 14(2), 157–163

work page 1976

[13] [13]

Stein, E. M. (1970). Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63). Princeton University Press. (NA) Department of Mathematics, Universiy of California Ir vine, Ir vine, CA 92697, USA Email address: nalpay@uci.edu (PI) Department of Mathematics, Universiy of California Ir vine, Ir vine, CA 92697, USA Email address: pivanisv@uci.edu

work page 1970