$L^q$ dimensions of self-similar measures, and applications: a survey

Pablo Shmerkin

arxiv: 1907.07121 · v1 · pith:EA3AQSJQnew · submitted 2019-07-13 · 🧮 math.DS · math.CA

L^q dimensions of self-similar measures, and applications: a survey

Pablo Shmerkin This is my paper

Pith reviewed 2026-05-24 21:40 UTC · model grok-4.3

classification 🧮 math.DS math.CA

keywords self-similar measuresL^q dimensionsexponential separationBernoulli convolutionsCantor set intersectionsFurstenberg conjecture

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The pith

A formula computes the L^q dimensions of self-similar measures on the real line under exponential separation.

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

The paper gives a self-contained proof of an explicit formula for the L^q dimensions of self-similar measures supported on the real line. The formula applies when the measures satisfy the exponential separation condition, relying on an inverse theorem for the L^q norm of convolutions as a black box. The same result is applied to obtain information about Bernoulli convolutions and intersections of self-similar Cantor sets.

Core claim

Under exponential separation, the L^q dimensions of self-similar measures on the real line equal a specific expression determined by the contraction ratios and probability weights, proved directly in one dimension as a simplified case of a prior general theorem, with the inverse convolution theorem taken as given.

What carries the argument

The exponential separation condition, which controls the spacing of the images under the iterated function system and enables the dimension formula.

If this is right

The L^q dimensions become computable from the defining parameters of the measure without further analysis.
New dimension bounds follow for Bernoulli convolutions that meet the separation hypothesis.
Dimension statements are obtained for intersections of self-similar Cantor sets on the line.

Where Pith is reading between the lines

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

The separation condition may serve as a template for analogous results on measures generated by other iterated function systems.
The one-dimensional proof structure could guide extensions to measures invariant under non-linear maps if suitable separation notions are identified.

Load-bearing premise

The measures obey the exponential separation condition and the inverse theorem for the L^q norm of convolutions holds.

What would settle it

A concrete self-similar measure on the line that satisfies exponential separation yet whose L^q dimensions deviate from the stated formula.

read the original abstract

We present a self-contained proof of a formula for the $L^q$ dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the $L^q$ norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.

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

This survey delivers a self-contained proof of the L^q dimension formula for self-similar measures on the line under exponential separation, but it is explicitly expository and adds no new theorems.

read the letter

Shmerkin has written a survey that gives a self-contained proof of the formula for L^q dimensions of self-similar measures on the real line when exponential separation holds. The argument treats the inverse theorem on L^q norms of convolutions as a black box from the 2019 Annals paper, which keeps the focus narrow and the presentation simpler than the general case. That choice makes sense for the stated goal of clarifying the reduction step in an important special case. The applications section then shows how the formula feeds into work on Bernoulli convolutions and intersections of self-similar Cantor sets, which is the part that will probably be most useful to readers already in the area. The paper does not claim to prove anything beyond what was already in the Annals article, and the abstract is upfront about the scope. The citation pattern is straightforward and does not create hidden circularity. The main limitation is simply that this is a survey rather than a research paper, so anyone who has already worked through the 2019 result will not find new mathematics here. The exposition appears aimed at making the ideas more accessible without reproving the heavy inverse theorem. This is the kind of piece that specialists in fractal geometry and dynamical systems might assign to students or use as a reference when they need the real-line case spelled out cleanly. I would send it for peer review at a journal that publishes surveys, because the self-contained proof and the concrete applications give it enough substance to justify referee time even though the core result is not new.

Referee Report

1 major / 2 minor

Summary. The paper presents a self-contained proof of the formula for the L^q dimensions of self-similar measures on the real line under the exponential separation condition (up to the inverse theorem for L^q norms of convolutions, treated as a black box from the author's 2019 Annals paper). It is framed as a simplification of the general result in a restricted but important setting, and includes a review of applications to Bernoulli convolutions and intersections of self-similar Cantor sets.

Significance. If the reduction holds, the survey strengthens accessibility to the core techniques by isolating the real-line exponential separation case, providing a clearer path to the dimension formula without the full generality of the 2019 result. Explicit applications to Bernoulli convolutions and Cantor set intersections connect the dimension estimates to concrete problems in fractal geometry, and the transparent treatment of the inverse theorem as a black box is a positive feature for readers.

major comments (1)

[Abstract] Abstract: the self-contained claim is qualified by excluding the inverse theorem, but the manuscript must explicitly identify the precise point in the proof (e.g., the convolution estimate or dimension reduction step) where the black-box result is invoked, so that the scope of the self-contained portion is unambiguous.

minor comments (2)

The applications section should state the precise open questions (e.g., exact dimension of Bernoulli convolutions for algebraic parameters) that the L^q formula resolves or advances.
Notation for the exponential separation condition should be recalled in the statement of the main theorem for readers who consult only the survey.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comment on clarifying the scope of the self-contained argument. We address the point below.

read point-by-point responses

Referee: [Abstract] Abstract: the self-contained claim is qualified by excluding the inverse theorem, but the manuscript must explicitly identify the precise point in the proof (e.g., the convolution estimate or dimension reduction step) where the black-box result is invoked, so that the scope of the self-contained portion is unambiguous.

Authors: We agree that an explicit identification of the invocation point will make the boundary of the self-contained portion unambiguous. In the revised manuscript we will add a short paragraph immediately after the statement of the main theorem (in the introduction) stating that the inverse theorem for L^q norms of convolutions is invoked precisely in the convolution estimate step that reduces the L^q dimension calculation to an application of the exponential separation hypothesis (corresponding to the passage from equation (3.2) to (3.3) in the proof). This will delineate the self-contained material without altering the overall structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained special-case proof with acknowledged black-box citation

full rationale

The paper explicitly frames its main contribution as a self-contained proof of the L^q dimension formula for self-similar measures on the line under exponential separation, up to an inverse theorem for L^q convolution norms treated as a black box from the author's prior Annals paper. This limitation is stated in the abstract, and the survey's goal is simplification of the reduction rather than reproof of the inverse theorem. No steps reduce by construction, self-definition, or renaming; the derivation chain is independent of the present paper's fitted values or internal inputs. Self-citation exists but is not load-bearing for the restricted result presented here.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects standard background assumptions in the field rather than new postulates.

axioms (1)

standard math Standard results from ergodic theory and harmonic analysis on self-similar measures and convolutions
The survey relies on established theory in dynamical systems and analysis.

pith-pipeline@v0.9.0 · 5649 in / 1136 out tokens · 29234 ms · 2026-05-24T21:40:31.407682+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.1: homogeneous WIFS with exponential separation implies D_μ(q) = min(dim_S(μ,q),1) via inverse theorem for ||ρ ∗ μ^(m)||_q
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.1 and Thm 2.2: (D,ℓ,R)-regular sets with branching R_s =1 or ≥2(1-δ)D on scale sets S

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 2 internal anchors

[1]

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

Christoph Aistleitner, Gerhard Larcher, and Mark Lewko . Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. Israel J. Math. , 222(1):463– 485, 2017. With an appendix by Jean Bourgain. 7

work page 2017
[2]

On the Hausdorff Dimension of Bernoulli Convolutions

Shigeki Akiyama, De-Jun Feng, T om Kempton, and T omas Per sson. On the Hausdorff di- mension of Bernoulli convolutions. Int. Math. Res. Not. IMRN , accepted for publication, 2018. arXiv:1801.07118. 29

work page internal anchor Pith review Pith/arXiv arXiv 2018
[3]

Dimension of slices ofSierpi ´ nski-like carpets.J

Bal´ azs B´ ar´ any and Michał Rams. Dimension of slices ofSierpi ´ nski-like carpets.J. Fractal Geom. , 1(3):273–294, 2014. 32

work page 2014
[4]

On the dimension of triangular self- afﬁne sets

BAL ´AZS B ´AR ´ANY , MICHAŁRAMS, and K´AROLY SIMON. On the dimension of triangular self- afﬁne sets. Ergodic Theory Dynam. Systems , 39(7):1751–1783, 2019. 6

work page 2019
[5]

V arj ´ u

Emmanuel Breuillard and P´ eter P . V arj ´ u. Entropy of Bernoulli convolutions and uniform expo- nential growth for linear groups. J. Anal. Math., accepted for publication, 2018. arXiv:1510.04043. 29

work page arXiv 2018
[6]

V arj ´ u

Emmanuel Breuillard and P´ eter P . V arj ´ u. On the dimensi on of Bernoulli convolutions. Ann. Probab., 47(4):2582–2617, 2019. 29

work page 2019
[7]

Spectral gaps, additive energy , and a fractal uncertainty prin- ciple

Semyon Dyatlov and Joshua Zahl. Spectral gaps, additive energy , and a fractal uncertainty prin- ciple. Geom. Funct. Anal., 26(4):1011–1094, 2016. 7

work page 2016
[8]

On a family of symmetric Bernoulli convolut ions

Paul Erd˝ os. On a family of symmetric Bernoulli convolut ions. Amer. J. Math. , 61:974–976, 1939. 28

work page 1939
[9]

On the smoothness properties of a family of B ernoulli convolutions

Paul Erd˝ os. On the smoothness properties of a family of B ernoulli convolutions. Amer. J. Math., 62:180–186, 1940. 27, 28

work page 1940
[10]

Fraser and Thomas Jordan

Jonathan M. Fraser and Thomas Jordan. The Assouad dimen sion of self-afﬁne carpets with no grid structure. Proc. Amer. Math. Soc., 145(11):4905–4918, 2017. 6

work page 2017
[11]

Adriano M. Garsia. Arithmetic properties of Bernoulli convolutions. T rans. Amer. Math. Soc. , 102:409–432, 1962. 28

work page 1962
[12]

On self-similar sets with overlaps an d inverse theorems for entropy

Michael Hochman. On self-similar sets with overlaps an d inverse theorems for entropy . Ann. of Math. (2), 180(2):773–822, 2014. 3, 7, 11, 12, 17, 21, 24, 25, 28, 29, 31

work page 2014
[13]

On self-similar sets with overlaps an d inverse theorems for entropy in Rd

Michael Hochman. On self-similar sets with overlaps an d inverse theorems for entropy in Rd. Mem. Amer. Math. Soc., 2017. 3, 24

work page 2017
[14]

Multifractal measures an d a weak separation condition

Ka-Sing Lau and Sze-Man Ngai. Multifractal measures an d a weak separation condition. Adv. Math., 141(1):45–96, 1999. 14

work page 1999
[15]

Existence of Lq dimensions and entropy dimension for self- conformal measures

Yuval Peres and Boris Solomyak. Existence of Lq dimensions and entropy dimension for self- conformal measures. Indiana Univ. Math. J., 49(4):1603–1621, 2000. 16

work page 2000
[16]

Problems on self-simil ar sets and self-afﬁne sets: an update

Yuval Peres and Boris Solomyak. Problems on self-simil ar sets and self-afﬁne sets: an update. In Fractal geometry and stochastics, II (Greifswald/Koserow , 1998), volume 46 of Progr. Probab., pages 95–106. Birkh¨ auser, Basel, 2000.3

work page 1998
[17]

On measures that improve $L^q$ dimension under convolution

Eino Rossi and Pablo Shmerkin. On measures that improve Lq dimension under convolution. Preprint, arXiv:1812.05660, 2018. 6, 10 Lq DIMENSIONS OF SELF-SIMILAR MEASURES 33

work page internal anchor Pith review Pith/arXiv arXiv 2018
[18]

On the exceptional set for absolute con tinuity of Bernoulli convolutions

Pablo Shmerkin. On the exceptional set for absolute con tinuity of Bernoulli convolutions. Geom. Funct. Anal., 24(3):946–958, 2014. 6, 26, 27, 28

work page 2014
[19]

On Furstenberg’s intersection conjec ture, self-similar measures, and the Lq norms of convolutions

Pablo Shmerkin. On Furstenberg’s intersection conjec ture, self-similar measures, and the Lq norms of convolutions. Ann. of Math. (2) , 189(2):319–391, 2019. 1, 2, 5, 6, 10, 13, 23, 30

work page 2019
[20]

Absolute continuit y of self-similar measures, their projec- tions and convolutions

Pablo Shmerkin and Boris Solomyak. Absolute continuit y of self-similar measures, their projec- tions and convolutions. T rans. Amer. Math. Soc., 368(7):5125–5151, 2016. 5, 26, 28

work page 2016
[21]

On the random series ∑± λn (an Erd˝ os problem).Ann

Boris Solomyak. On the random series ∑± λn (an Erd˝ os problem).Ann. of Math. (2) , 142(3):611– 625, 1995. 28

work page 1995
[22]

On the dimension of Bernoulli convolutions for all transcendental parameters

P´ eter V arj ´ u. On the dimension of Bernoulli convolutions for all transcendental parameters. Ann. of Math. (2) , 189(3):1001–1011, 2019. 29

work page 2019
[23]

V arj ´ u

P´ eter P . V arj ´ u. Absolute continuity of Bernoulli convolutions for algebraic parameters. J. Amer. Math. Soc., 32(2):351–397, 2019. 28 DEPARTMENT OF MATHEMATICS AND STATISTICS , T ORCUATO DI TELLA UNIVERSITY , AND CON- ICET, B UENOS AIRES , A RGENTINA E-mail address: pshmerkin@utdt.edu URL: http://www.utdt.edu/profesores/pshmerkin

work page 2019

[1] [1]

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

Christoph Aistleitner, Gerhard Larcher, and Mark Lewko . Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. Israel J. Math. , 222(1):463– 485, 2017. With an appendix by Jean Bourgain. 7

work page 2017

[2] [2]

On the Hausdorff Dimension of Bernoulli Convolutions

Shigeki Akiyama, De-Jun Feng, T om Kempton, and T omas Per sson. On the Hausdorff di- mension of Bernoulli convolutions. Int. Math. Res. Not. IMRN , accepted for publication, 2018. arXiv:1801.07118. 29

work page internal anchor Pith review Pith/arXiv arXiv 2018

[3] [3]

Dimension of slices ofSierpi ´ nski-like carpets.J

Bal´ azs B´ ar´ any and Michał Rams. Dimension of slices ofSierpi ´ nski-like carpets.J. Fractal Geom. , 1(3):273–294, 2014. 32

work page 2014

[4] [4]

On the dimension of triangular self- afﬁne sets

BAL ´AZS B ´AR ´ANY , MICHAŁRAMS, and K´AROLY SIMON. On the dimension of triangular self- afﬁne sets. Ergodic Theory Dynam. Systems , 39(7):1751–1783, 2019. 6

work page 2019

[5] [5]

V arj ´ u

Emmanuel Breuillard and P´ eter P . V arj ´ u. Entropy of Bernoulli convolutions and uniform expo- nential growth for linear groups. J. Anal. Math., accepted for publication, 2018. arXiv:1510.04043. 29

work page arXiv 2018

[6] [6]

V arj ´ u

Emmanuel Breuillard and P´ eter P . V arj ´ u. On the dimensi on of Bernoulli convolutions. Ann. Probab., 47(4):2582–2617, 2019. 29

work page 2019

[7] [7]

Spectral gaps, additive energy , and a fractal uncertainty prin- ciple

Semyon Dyatlov and Joshua Zahl. Spectral gaps, additive energy , and a fractal uncertainty prin- ciple. Geom. Funct. Anal., 26(4):1011–1094, 2016. 7

work page 2016

[8] [8]

On a family of symmetric Bernoulli convolut ions

Paul Erd˝ os. On a family of symmetric Bernoulli convolut ions. Amer. J. Math. , 61:974–976, 1939. 28

work page 1939

[9] [9]

On the smoothness properties of a family of B ernoulli convolutions

Paul Erd˝ os. On the smoothness properties of a family of B ernoulli convolutions. Amer. J. Math., 62:180–186, 1940. 27, 28

work page 1940

[10] [10]

Fraser and Thomas Jordan

Jonathan M. Fraser and Thomas Jordan. The Assouad dimen sion of self-afﬁne carpets with no grid structure. Proc. Amer. Math. Soc., 145(11):4905–4918, 2017. 6

work page 2017

[11] [11]

Adriano M. Garsia. Arithmetic properties of Bernoulli convolutions. T rans. Amer. Math. Soc. , 102:409–432, 1962. 28

work page 1962

[12] [12]

On self-similar sets with overlaps an d inverse theorems for entropy

Michael Hochman. On self-similar sets with overlaps an d inverse theorems for entropy . Ann. of Math. (2), 180(2):773–822, 2014. 3, 7, 11, 12, 17, 21, 24, 25, 28, 29, 31

work page 2014

[13] [13]

On self-similar sets with overlaps an d inverse theorems for entropy in Rd

Michael Hochman. On self-similar sets with overlaps an d inverse theorems for entropy in Rd. Mem. Amer. Math. Soc., 2017. 3, 24

work page 2017

[14] [14]

Multifractal measures an d a weak separation condition

Ka-Sing Lau and Sze-Man Ngai. Multifractal measures an d a weak separation condition. Adv. Math., 141(1):45–96, 1999. 14

work page 1999

[15] [15]

Existence of Lq dimensions and entropy dimension for self- conformal measures

Yuval Peres and Boris Solomyak. Existence of Lq dimensions and entropy dimension for self- conformal measures. Indiana Univ. Math. J., 49(4):1603–1621, 2000. 16

work page 2000

[16] [16]

Problems on self-simil ar sets and self-afﬁne sets: an update

Yuval Peres and Boris Solomyak. Problems on self-simil ar sets and self-afﬁne sets: an update. In Fractal geometry and stochastics, II (Greifswald/Koserow , 1998), volume 46 of Progr. Probab., pages 95–106. Birkh¨ auser, Basel, 2000.3

work page 1998

[17] [17]

On measures that improve $L^q$ dimension under convolution

Eino Rossi and Pablo Shmerkin. On measures that improve Lq dimension under convolution. Preprint, arXiv:1812.05660, 2018. 6, 10 Lq DIMENSIONS OF SELF-SIMILAR MEASURES 33

work page internal anchor Pith review Pith/arXiv arXiv 2018

[18] [18]

On the exceptional set for absolute con tinuity of Bernoulli convolutions

Pablo Shmerkin. On the exceptional set for absolute con tinuity of Bernoulli convolutions. Geom. Funct. Anal., 24(3):946–958, 2014. 6, 26, 27, 28

work page 2014

[19] [19]

On Furstenberg’s intersection conjec ture, self-similar measures, and the Lq norms of convolutions

Pablo Shmerkin. On Furstenberg’s intersection conjec ture, self-similar measures, and the Lq norms of convolutions. Ann. of Math. (2) , 189(2):319–391, 2019. 1, 2, 5, 6, 10, 13, 23, 30

work page 2019

[20] [20]

Absolute continuit y of self-similar measures, their projec- tions and convolutions

Pablo Shmerkin and Boris Solomyak. Absolute continuit y of self-similar measures, their projec- tions and convolutions. T rans. Amer. Math. Soc., 368(7):5125–5151, 2016. 5, 26, 28

work page 2016

[21] [21]

On the random series ∑± λn (an Erd˝ os problem).Ann

Boris Solomyak. On the random series ∑± λn (an Erd˝ os problem).Ann. of Math. (2) , 142(3):611– 625, 1995. 28

work page 1995

[22] [22]

On the dimension of Bernoulli convolutions for all transcendental parameters

P´ eter V arj ´ u. On the dimension of Bernoulli convolutions for all transcendental parameters. Ann. of Math. (2) , 189(3):1001–1011, 2019. 29

work page 2019

[23] [23]

V arj ´ u

P´ eter P . V arj ´ u. Absolute continuity of Bernoulli convolutions for algebraic parameters. J. Amer. Math. Soc., 32(2):351–397, 2019. 28 DEPARTMENT OF MATHEMATICS AND STATISTICS , T ORCUATO DI TELLA UNIVERSITY , AND CON- ICET, B UENOS AIRES , A RGENTINA E-mail address: pshmerkin@utdt.edu URL: http://www.utdt.edu/profesores/pshmerkin

work page 2019