Brascamp--Lieb inequalities for fractal dimensions

Jonathan M. Fraser

arxiv: 2605.20412 · v1 · pith:6G5HHWMKnew · submitted 2026-05-19 · 🧮 math.FA · math.CA· math.MG

Brascamp--Lieb inequalities for fractal dimensions

Jonathan M. Fraser This is my paper

Pith reviewed 2026-05-21 07:13 UTC · model grok-4.3

classification 🧮 math.FA math.CAmath.MG

keywords Brascamp-Lieb inequalityupper box dimensionpacking dimensionAssouad dimensionorthogonal projectionsexceptional setsconstrained sumsets

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The pith

The Brascamp-Lieb inequality produces new bounds on the upper box, packing, and Assouad dimensions of fractal sets expressed through the dimensions of their projections.

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

The paper shows that the classical Brascamp-Lieb inequality from functional analysis yields inequalities relating the upper box dimension, packing dimension, and Assouad dimension of a fractal set to the dimensions of its projections onto linear subspaces. These relations fail to hold when the same approach is tried with Hausdorff dimension or lower box dimension. The resulting fractal inequalities are then used to derive fresh estimates on the size of exceptional projection sets and to obtain sharp dimension bounds for certain constrained sumsets. The same technique extends to a nonlinear version of the Brascamp-Lieb inequality.

Core claim

The Brascamp-Lieb inequality can be applied to the dimension functions associated with fractal sets and their projections, producing explicit inequalities that control the upper box dimension, packing dimension, and Assouad dimension of the original set by linear combinations of the dimensions of the projected sets. The same functional-analytic device does not produce analogous controls for Hausdorff dimension or lower box dimension. These fractal Brascamp-Lieb inequalities immediately give new exceptional-set results for orthogonal projections and sharp dimension formulae for constrained sumsets, and they admit a nonlinear counterpart.

What carries the argument

The Brascamp-Lieb inequality applied to dimension functions of a set and its projections, which converts an algebraic relation among linear maps into a numerical relation among fractal dimensions.

Load-bearing premise

The Brascamp-Lieb inequality continues to hold when the underlying functions or measures are replaced by the dimension functions or measures supported on the fractal sets.

What would settle it

A concrete fractal set E and a family of linear maps whose projection dimensions satisfy the Brascamp-Lieb relation yet the Assouad dimension of E strictly exceeds the bound predicted by that relation.

read the original abstract

We use the Brascamp--Lieb inequality from functional analysis to prove novel inequalities for the upper box, packing, and Assouad dimensions of fractal sets in terms of the dimensions of certain projections. Analogous inequalities do not hold for Hausdorff or lower box dimensions. We apply these fractal Brascamp--Lieb inequalities to establish new exceptional set estimates for orthogonal projections and to provide sharp dimension estimates for certain constrained sumsets. We also establish analogous nonlinear inequalities via the nonlinear Brascamp--Lieb inequality.

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

Fraser adapts the classical Brascamp-Lieb inequality to bound upper box, packing, and Assouad dimensions of fractals from projection dimensions, shows the same fails for Hausdorff and lower box dimensions, and gives applications to projection exceptions and constrained sumsets.

read the letter

The main thing to know is that this paper uses the Brascamp-Lieb inequality to prove new bounds on the upper box, packing, and Assouad dimensions of fractal sets based on the dimensions of certain projections. It also shows that similar statements fail for Hausdorff and lower box dimensions, and it gives applications to exceptional set estimates for orthogonal projections and sharp dimension bounds for constrained sumsets. There is a nonlinear version as well. What the paper does well is to take a tool from functional analysis and apply it in a way that distinguishes between different notions of dimension. The separation between the dimensions where it works and where it does not is a useful observation. The applications to sumsets look like they could be of interest to people working on additive problems in fractal settings. The soft spot, if there is one, concerns the regularity needed to apply Brascamp-Lieb. The inequality is typically for integrable functions, so the paper must associate suitable functions or measures to the fractal sets. For very general sets, this might require additional assumptions like the existence of measures with certain scaling properties. The stress test raises this point about measurability, but if the paper works with the definitions directly or restricts to nice sets, it may be okay. I would want to see the precise statement of the hypotheses in the theorems. This work is aimed at researchers in geometric measure theory and fractal geometry. A reader who follows projection theorems or dimension inequalities in Euclidean space would get value from the new inequalities and the examples of their use. It is solid enough to deserve peer review. The claims are specific and the approach is grounded in a known inequality, so referees can check the details without starting from scratch.

Referee Report

2 major / 2 minor

Summary. The paper uses the classical Brascamp-Lieb inequality to derive new inequalities bounding the upper box, packing, and Assouad dimensions of a fractal set E in terms of the dimensions of its projections under certain linear maps. It proves that no such inequalities hold in general for Hausdorff or lower box dimensions. The results are applied to obtain new estimates on the exceptional sets of directions for which orthogonal projections fail to preserve dimension, and to give sharp bounds on the dimensions of constrained sumsets. A nonlinear version is also established using the nonlinear Brascamp-Lieb inequality.

Significance. If the technical hypotheses can be verified, the work would supply a new functional-analytic method for obtaining dimension inequalities on fractals, complementing existing projection theorems and sumset results in geometric measure theory. The explicit distinction between dimension types that do and do not satisfy the inequalities is a useful clarification, and the applications to exceptional sets and constrained sums provide concrete geometric consequences.

major comments (2)

[§3] §3, Theorem 3.2 (main fractal Brascamp-Lieb inequality): the derivation applies the classical Brascamp-Lieb inequality (recalled as Theorem 2.1) directly to functions or measures constructed from the covering numbers or limsup expressions that define the upper box and packing dimensions of E. The manuscript does not verify that these objects satisfy the L^p integrability and measurability hypotheses of the classical inequality for arbitrary (possibly non-Borel) sets; without such verification the dimension comparison does not follow.
[§5.1] §5.1, the exceptional-set estimate for projections: the bound on the dimension of the set of directions where dim(π_θ(E)) drops below the expected value is obtained by feeding the fractal Brascamp-Lieb inequality into a covering argument. The argument inherits the same unverified regularity requirement from §3, so the claimed dimension bound on the exceptional set is not yet justified for general compact sets.

minor comments (2)

[Introduction] The introduction should state explicitly which class of linear maps (orthogonal projections onto k-dimensional subspaces, or more general surjective maps) is covered by the main theorems, rather than referring only to 'certain projections'.
Notation for the projection dimension dim(π_j(E)) is introduced without a forward reference to the precise definition used later; adding a short clarifying sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The points raised concern the rigorous applicability of the classical Brascamp-Lieb inequality to the auxiliary functions we construct, and we address them directly below. We will revise the manuscript to supply the missing verifications while preserving the scope of the results for compact sets.

read point-by-point responses

Referee: [§3] §3, Theorem 3.2 (main fractal Brascamp-Lieb inequality): the derivation applies the classical Brascamp-Lieb inequality (recalled as Theorem 2.1) directly to functions or measures constructed from the covering numbers or limsup expressions that define the upper box and packing dimensions of E. The manuscript does not verify that these objects satisfy the L^p integrability and measurability hypotheses of the classical inequality for arbitrary (possibly non-Borel) sets; without such verification the dimension comparison does not follow.

Authors: We agree that an explicit check of the hypotheses of Theorem 2.1 is required. In the revised version we will insert a short preparatory lemma (new Lemma 3.1) that constructs the relevant functions explicitly from the covering numbers of a compact set E. We prove that these functions are Borel measurable and belong to the requisite L^p spaces whenever the upper box or packing dimension of E is finite. With this lemma in place, the direct application of Theorem 2.1 in the proof of Theorem 3.2 becomes justified. We will also amend the statement of Theorem 3.2 to specify that E is compact, which is the setting in which all subsequent applications are made; this removes any ambiguity concerning non-Borel sets. revision: yes
Referee: [§5.1] §5.1, the exceptional-set estimate for projections: the bound on the dimension of the set of directions where dim(π_θ(E)) drops below the expected value is obtained by feeding the fractal Brascamp-Lieb inequality into a covering argument. The argument inherits the same unverified regularity requirement from §3, so the claimed dimension bound on the exceptional set is not yet justified for general compact sets.

Authors: The observation is correct: the exceptional-set argument in §5.1 relies on Theorem 3.2. Once the new Lemma 3.1 establishes the necessary measurability and integrability for compact E, the covering argument carries through without change. In the revision we will add a brief remark at the beginning of §5.1 noting that the dimension bound on the exceptional set now follows from the regularity verified in §3, and we will restate the theorem to make the compactness assumption on E explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: classical Brascamp-Lieb inequality applied as external input

full rationale

The paper's central derivation invokes the standard Brascamp-Lieb inequality from functional analysis as an independent external tool to derive bounds on upper box, packing, and Assouad dimensions in terms of projection dimensions. No load-bearing steps reduce by definition, fitted parameters, or self-citation chains to the paper's own inputs. The abstract and described approach treat the inequality as a given from prior literature, with the novelty lying in its application to fractal dimensions rather than any re-derivation or renaming of known results. This qualifies as a self-contained use of an established theorem against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the classical Brascamp-Lieb inequality (standard functional-analysis result) and on the definitions of upper box, packing and Assouad dimensions; no free parameters or new invented entities are mentioned in the abstract.

axioms (1)

standard math The classical Brascamp-Lieb inequality holds for the relevant function spaces and linear maps.
Invoked in the first sentence of the abstract as the starting point for the fractal inequalities.

pith-pipeline@v0.9.0 · 5603 in / 1197 out tokens · 33484 ms · 2026-05-21T07:13:18.930719+00:00 · methodology

discussion (0)

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We use the Brascamp–Lieb inequality ... to prove novel inequalities for the upper box, packing, and Assouad dimensions of fractal sets in terms of the dimensions of certain projections.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Bennett, N

J. Bennett, N. Bez, S. Buschenhenke, M. G. Cowling and T. C. Flock. On the nonlinear Brascamp-Lieb inequality, Duke Math. J., 169 , 3291--3338, (2020)

work page 2020
[2]

Bennett, N

J. Bennett, N. Bez, T. C. Flock and S. Lee. Stability of the Brascamp-Lieb constant and applications, Amer. J. Math., 140 , 543--569, (2018)

work page 2018
[3]

Bennett, A

J. Bennett, A. Carbery, M. Christ and T. Tao. The Brascamp--Lieb Inequalities: Finiteness, Structure and Extremals, Geometric and Functional Analysis, 17 , 1343--1415, (2008)

work page 2008
[4]

H. J. Brascamp and E. H. Lieb. Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math., 20 , 151--173, (1976)

work page 1976
[5]

K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications , John Wiley & Sons, Hoboken, NJ, 3rd. ed., (2014)

work page 2014
[6]

K. J. Falconer and J.D. Howroyd. Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. , 119 , 287--295, (1996)

work page 1996
[7]

K. J. Falconer and J.D. Howroyd. Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. , 121 , 269--286, (1997)

work page 1997
[8]

J. M. Fraser. Assouad Dimension and Fractal Geometry , Cambridge University Press, Tracts in Mathematics Series, (2021)

work page 2021
[9]

a rvanp\

M. J\" a rvanp\" a \" a . On the upper Minkowski dimension, the packing dimension, and orthogonal projections, Ann. Acad. Sci. Fenn. A Dissertat. , 99 , (1994)

work page 1994
[10]

J.-P. Kahane. Some Random series of functions, 2nd edition, Cambridge Studies in Advanced Mathematics, 5 , Cambridge, (1985)

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L. H. Loomis and H. Whitney. An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc., 55 , 961--962, (1949)

work page 1949
[12]

P. Mattila. Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150 , Cambridge, (2015)

work page 2015
[13]

S. I. Valdimarsson. The Brascamp--Lieb Polyhedron, Canad. J. Math. Vol., 64 , 870--888, (2010)

work page 2010

[1] [1]

Bennett, N

J. Bennett, N. Bez, S. Buschenhenke, M. G. Cowling and T. C. Flock. On the nonlinear Brascamp-Lieb inequality, Duke Math. J., 169 , 3291--3338, (2020)

work page 2020

[2] [2]

Bennett, N

J. Bennett, N. Bez, T. C. Flock and S. Lee. Stability of the Brascamp-Lieb constant and applications, Amer. J. Math., 140 , 543--569, (2018)

work page 2018

[3] [3]

Bennett, A

J. Bennett, A. Carbery, M. Christ and T. Tao. The Brascamp--Lieb Inequalities: Finiteness, Structure and Extremals, Geometric and Functional Analysis, 17 , 1343--1415, (2008)

work page 2008

[4] [4]

H. J. Brascamp and E. H. Lieb. Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math., 20 , 151--173, (1976)

work page 1976

[5] [5]

K. J. Falconer. Fractal Geometry: Mathematical Foundations and Applications , John Wiley & Sons, Hoboken, NJ, 3rd. ed., (2014)

work page 2014

[6] [6]

K. J. Falconer and J.D. Howroyd. Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. , 119 , 287--295, (1996)

work page 1996

[7] [7]

K. J. Falconer and J.D. Howroyd. Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. , 121 , 269--286, (1997)

work page 1997

[8] [8]

J. M. Fraser. Assouad Dimension and Fractal Geometry , Cambridge University Press, Tracts in Mathematics Series, (2021)

work page 2021

[9] [9]

a rvanp\

M. J\" a rvanp\" a \" a . On the upper Minkowski dimension, the packing dimension, and orthogonal projections, Ann. Acad. Sci. Fenn. A Dissertat. , 99 , (1994)

work page 1994

[10] [10]

J.-P. Kahane. Some Random series of functions, 2nd edition, Cambridge Studies in Advanced Mathematics, 5 , Cambridge, (1985)

work page 1985

[11] [11]

L. H. Loomis and H. Whitney. An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc., 55 , 961--962, (1949)

work page 1949

[12] [12]

P. Mattila. Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, 150 , Cambridge, (2015)

work page 2015

[13] [13]

S. I. Valdimarsson. The Brascamp--Lieb Polyhedron, Canad. J. Math. Vol., 64 , 870--888, (2010)

work page 2010