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Smooth projections of self-similar measures

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Pith's one-line read This paper proves that, in dimensions at least three, a prescribed projection of a self-similar measure is absolutely continuous with a Besov-class density whenever the rotations mix exponentially fast compared with the measure's orbit-rela

desk verdict First fully explicit criterion for smoothness of prescribed projections of self-similar measures, with striking constructions from Ramanujan sets; the core proofs are sound, though the LPS equality (1.3) should be weakened to the upper bound it actually supports. read the letter →

arxiv 2607.15635 v1 pith:JU4HZMYH submitted 2026-07-17 math.DS math.CA

Amir Algom , Federico Rodriguez Hertz , Zhiren Wang This is my paper · ORCID
classification math.DSmath.CA MSC 28A8042B1037C45
keywords self-similarmeasuresorthogonalprojectionsabsolutecontinuityspectralgapFourierdimensionBesovspacessphericalaveragesRamanujansets
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

Self-similar measures in R^d, d≥3, can be certified projection by projection: the paper proves that a given orthogonal projection is absolutely continuous, with a quantified Besov regularity of its density, as soon as the rotational part of the iterated function system mixes exponentially fast at a rate that beats the measure's dimension relative to that projection's orbit. The theorem reduces this to an inequality between the spectral gap of an averaging operator on the orbit and a relative L^q dimension, and the proof is Fourier-analytic: it decomposes the projected Fourier transform into dyadic annuli, runs the self-similarity through a random walk on rotated directions, and controls the error by Sobolev embedding on the orbit. Using explicitly constructed rotation sets with optimal spectral gap, the paper produces singular self-similar measures whose every line projection is absolutely continuous, measures of arbitrarily small Fourier dimension whose projections are smooth in all but a fully explicit exceptional set of directions, and a non-trivial self-similar measure that is Salem (Fourier dimension equals Hausdorff dimension) with a C^2_0 density. This matters because it is the first general and fully explicit criterion for smoothness of a prescribed projection, and it shows that singular ambient measures can still have remarkably smooth projection structure.

What carries the argument

The central object is the averaging operator P_{a,V} on the compact homogeneous orbit O_V of the prescribed subspace under the rotation group; its norm on the zero-mean L^2 space measures how fast rotational mixing erases the non-invariant part of any function on the orbit. The comparison quantity is the orbit-relative L^q dimension dim^V_q ν, defined by summability of orbit-averaged Littlewood-Paley norms of the projected measure; it is the integrability threshold of the spherical averages that the proof tries to dominate. The spectral inequality is the condition that the exponential mixing term decays faster than the polynomial growth of the annuli and of the Sobolev norm of the frequency-

What would settle it

Compute the operator norm on L^2_0(SO(3)) of the averaging operator over the explicit 18-element rotation set used in the R^3 example; if it exceeds 2√17/18, then the claimed absolute continuity of all line projections is not certified by the criterion, and if some line projection is actually singular, the criterion itself is false. More generally, for the R^4 family, evaluate both sides of the spectral inequality for a chosen prime p and dimension D; an inequality that holds while the projected measure fails to be in L^2 would refute the theorem.

Watch

Extended reading notes

Core claim

The central claim (Theorem 1.9) is a conditional smoothness criterion. For a self-similar IFS on R^d (d≥3) with measure ν, fix a k-plane V and let G be the closed rotation group with orbit O_V=G·V. For each contraction modulus a, let P_{a,V} be the averaging operator on the zero-mean L^2 space of O_V, and let dim^V_q ν be the L^q dimension of ν relative to O_V. If the spectral ratio Λ = [−log Σ_a r_{p_a}‖P_{a,V}‖] / max_i(−log|r_i|) exceeds (b(q−1)+b(σ+b(q−1)))/(S−b) for some k<b<S<dim^V_q ν (with σ>½ dim O_V and, when q<2, σ<q), then π_V ν is absolutely continuous and its density lies in the Besov space B^{(b−k)/q′}_{q,q}(V), hence in L^q(V).

Load-bearing premise

For the explicit applications, everything rests on the exact spectral norm of the averaging operator on the optimally mixing rotation set: if that norm were even slightly larger than the stated value, the numerical inequalities that drive the examples would no longer hold.

Editorial extensions

If this is right

  • For any self-similar measure satisfying the spectral inequality, every prescribed projection is not merely dimension-preserving but has an L^q density with explicit Besov regularity; the certificate is a computation of an operator norm and a relative dimension.
  • Singular self-similar measures on R^3 exist whose every line projection is absolutely continuous, so singularity of the ambient measure is no obstruction to smoothness of all its line projections.
  • On R^4, for any dimension D between (3+√21)/2 and 4, there is a singular self-similar measure of dimension D all of whose line projections have L^2 densities.
  • The phenomenon is independent of Fourier dimension: for any η>0 one can force the Fourier dimension below η while keeping all projections onto planes not containing a fixed direction smooth up to Sobolev order (4−k)/2−τ.
  • There exist non-trivial self-similar measures on R^3 that are Salem (Fourier dimension equals Hausdorff dimension) and have a C^2_0 density with non-empty interior of the support.

Reading between the lines

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

  • The spectral inequality probably admits a sharper form in which the threshold depends on the whole Lyapunov spectrum; replacing the maximum contraction rate by the Lyapunov exponent, as the paper's remark does, suggests that generically the threshold is easier to satisfy, so one may expect many more explicit examples than the ones constructed.
  • A natural test is to push the construction toward the boundary: letting the dimension D approach 4 in the R^4 example should make the L^2 density approach C^1 regularity, indicating a critical threshold at which the density gains a full derivative.
  • The exceptional-direction mechanism in the small-Fourier-dimension example — a singular marginal along one coordinate forcing singularity of any projection containing that direction — suggests a general slicing principle: for product-like self-similar couplings, the set of singular projection directions is controlled by the most singular marginal, not by the full measure.
  • Because the criterion is fully explicit, it can be used as a computational certificate: for a given algebraic IFS one can numerically bound the operator norms and relative dimensions and verify or refute absolute continuity of a specific direction before any symbolic integration.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper proves a Furstenberg-type criterion for individual orthogonal projections of self-similar measures to be absolutely continuous with quantified Sobolev/Besov regularity. The main result, Theorem 1.9, says that if the rotational part of the IFS is exponentially mixing (spectral gap) on the orbit of the prescribed subspace V, and if the rate of mixing is fast compared with a certain orbit-relative L^q dimension of the measure, then the projected measure is absolutely continuous and its density lies in a Besov space. The proof combines a model decomposition for non-uniform contraction ratios, random-walk estimates on the group/orbit, Sobolev embedding, and Littlewood–Paley analysis. The abstract applications use Lubotzky–Phillips–Sarnak Ramanujan sets in SO(3): singular self-similar measures with every line projection absolutely continuous, measures with arbitrarily small Fourier dimension but smooth projections in all but one explicit direction, and a self-similar Salem measure with C^2 density. The paper is carefully written and the main mechanism is original, but the explicit applications rest on a false reading of the LPS theorem that needs correction.

Significance. If the results hold, this is the first general fully explicit criterion ensuring smoothness of a prescribed projection of a self-similar measure, and the applications are striking: they show that spectral gap can completely overcome Marstrand-type exceptional directions, even when the ambient measure is singular and has very small Fourier dimension. The paper also provides a clean Besov-space framework and an orbit-relative dimension that may be useful beyond this setting. The proofs are detailed and largely self-contained, and the numerical conditions are concrete and checkable. The main reservation concerns not the strategy but the incorrect use of the LPS spectral equality in the explicit applications; this is fixable and the applications survive because the actual direction of the inequality is favorable.

major comments (1)
  1. [§1.3, Eq. (1.3)] The statement that for every prime p≡1 mod 4 the LPS Ramanujan set satisfies ∥P_G∥_{L^2_0(SO(3))}=2√p/(p+1) is not correct. The LPS theorem gives an upper bound, and the bound need not be attained for each p. For p=5, the six rotations from quaternions of norm 5 all have rotation angle θ with cos θ=−3/5; on the spin-ℓ representation the averaging operator is scalar with eigenvalue χ_ℓ(θ)/(2ℓ+1), and the supremum over ℓ is ≈0.159, far below √5/3≈0.745. This is not a harmless citation slip: Sections 5.1, 5.2, 5.3 and 5.6 use Eq. (1.3) as an equality in the numerical verifications. In every use the inequality direction is favorable — replacing '=' by '≤' makes the spectral gap larger — so the proofs survive verbatim. Nevertheless, the manuscript must be corrected to state the LPS result as an upper bound, and the wording 'precisely'/'achieved' must be adjusted. Because the explicit applicat
minor comments (4)
  1. [§5.3, Corollary 1.5] The assertion dim_F μ_{r,ε} ≤ dim_F λ_ε is described as 'direct to check'. It is true, since the Fourier transform of the fourth-coordinate marginal equals the restriction of the ambient Fourier transform to the vertical frequency axis, but a one-sentence justification would help the reader, especially because this is used to conclude that the ambient Fourier dimension is small.
  2. [§5.1] In the numerical check, the expression 'log 18/(2√17)' is ambiguous. It should be written as −log(2√17/18) or with explicit parentheses.
  3. [§1.3 and §5.2] The notation R_p is used both for the set of rotations and for the averaging operator norm, and the dependence on p is sometimes omitted. Adding a subscript or explicit dependence would improve readability.
  4. [§3.2, Lemma 3.4] The set in the displayed inequality is written as 'sup (t0u ∪ {...})'; the braces are visually confusing. This is a minor typographical issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.9 is a genuine implication and the applications verify its hypotheses.

full rationale

The central result, Theorem 1.9, is a conditional implication proved in Section 4 from explicit hypotheses: a spectral-gap estimate on averaged rotational operators and a relative L^q-dimension condition k<b<S<dim^V_q ν. The conclusion is Besov regularity of the projected measure π_V ν. These are not identified with each other: the input is an orbit-averaged energy over U∈O_V, while the conclusion concerns the single prescribed subspace V. The proof uses Jensen, self-similarity, Sobolev embedding, and the spectral gap to replace the fixed-direction Littlewood-Paley term by the orbit average plus a controlled error; the orbit average is then bounded by E^V_{q,S}(ν)<∞. Thus the derivation does not reduce to the definition of dim^V_q ν or to the spectral-gap assumption by construction. The relative dimension is defined independently in Section 3, and although both it and Besov spaces use Littlewood-Paley kernels, the theorem explicitly separates the input exponent S from the target Besov exponent b. The applications (Corollaries 1.3–1.7) select explicit IFS data—Ramanujan rotation sets, translations, contraction ratios, weights—and verify the hypotheses of Theorems 1.2/1.8/1.9 using external results (LPS, Hochman, Cawley–Mauldin, Falconer, Shmerkin–Solomyak). Parameters are chosen to satisfy numerical inequalities, not fitted to the regularity conclusions. Self-citations [2,3,4,5] are used for strategy and model-decomposition inspiration, but all technical lemmas actually invoked (Lemma 2.1, Lemma 2.2, Claim 2.3, Proposition 2.5, Corollary 2.6) are proved within the paper, so no load-bearing self-citation is present. The reviewer-flagged issue that equality (1.3) may fail is a correctness concern about an externally cited input, not a circularity: every numerical use of (1.3) goes in the direction of an upper bound on the operator norm, and a smaller true norm only strengthens the spectral-gap inequalities. No uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via self-citation. Verdict: no significant circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new physical or mathematical entities are postulated. The 'orbit-relative L^q dimension' is a new definition but is a measurable quantity of the given measure, not an invented entity. The free parameters listed are explicit construction parameters, not hidden fitted constants. The axioms are external theorems or standard background facts; none are ad hoc to this paper.

free parameters (5)
  • contraction ratio r in Corollary 1.3 = 18^{-2/5}
    Chosen so the symbolic dimension is 5/2 and the LPS spectral inequality holds; not fitted to data.
  • D and prime p in Corollary 1.4 = D ∈ ((3+√21)/2, 4), p ≡ 1 mod 4 sufficiently large with explicit lower bounds
    Chosen to satisfy the SSC and the spectral inequality; the bounds are explicit in the statement.
  • r and ε in Corollary 1.5 = r close to 1 satisfying (5.7),(5.8); ε small satisfying (5.9)
    Selected so that the Fourier dimension is arbitrarily small and the spherical average contracts at rate >4.
  • q (weight) and r in Corollary 1.6 = q close to 1, r = exp(-a(q)/D)
    Chosen so dim_2 ν is close to 1 while projections remain smooth; explicit formulas for a(q).
  • contraction ratio r in Corollary 1.7 = 1977/2000
    Explicit rational chosen so the spherical average decays faster than any 2t+3 bound needed for H^t, t<3.9.
assumptions (6)
  • domain assumption LPS Ramanujan spectral bound ∥P_p∥_{L^2_0(SO(3))} = 2√p/(p+1) (Lubotzky–Phillips–Sarnak)
    External theorem used in all explicit corollaries, first at (1.3).
  • domain assumption Hochman's dimension formula for algebraic IFS with no exact overlaps [32, Cor. 1.7]
    Used to compute dimν=5/2 in Corollary 1.3.
  • domain assumption Exact dimensionality of self-similar measures and lim dim_q ν = dim ν (Feng–Hu, Peres–Solomyak, Shmerkin–Solomyak)
    Used to deduce Theorem 1.2 from Theorem 1.9 in Section 1.5.
  • standard math Peres–Schlag spherical-average energy identity [45, Prop. 2.2]
    Used in Lemma 3.1 and throughout the L^2 estimates.
  • standard math Sobolev embedding on compact homogeneous spaces, (2.5)
    Used in Claim 2.3, Corollary 2.6, and the proof of Theorem 1.9.
  • domain assumption Feng–Hu and Shmerkin–Solomyak results on L^q dimensions of self-similar measures
    Used in the deduction of Theorem 1.2 Part (1).

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Pith. "Pith review of Smooth projections of self-similar measures." pith.science (2026). https://pith.science/paper/JU4HZMYH

@misc{pith2026260715635,
  author       = {Pith},
  title        = {Pith review of: Smooth projections of self-similar measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JU4HZMYH}},
  note         = {Machine review of arXiv:2607.15635}
}
abstract

We prove a Furstenberg-type criterion for a given orthogonal projection of a self-similar measure to be absolutely continuous, with quantified regularity. It requires exponential mixing of the rotational part at a rate that is sufficiently fast compared with an orbit relative analogue of its dimension. Using Ramanujan sets of irrational rotations in \(\mathrm{SO}(3)\) constructed by Lubotzky, Phillips and Sarnak (1986, 1987), we obtain explicit applications. In particular, we exhibit singular self-similar measures whose every line projection is absolutely continuous, measures of arbitrarily small Fourier dimension with smooth projections in all but a fully explicit exceptional set of directions, and a non-trivial example of a self-similar measure that is Salem with a $C^2 _0$ density.

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Reference graph

Works this paper leans on

62 extracted references · 8 linked inside Pith

  1. [1]

    On normal numbers and self-similar measures.Adv

    Amir Algom, Simon Baker, and Pablo Shmerkin. On normal numbers and self-similar measures.Adv. Math., 399:Paper No. 108276, 17, 2022

    2022

  2. [2]

    Logarithmic Fourier decay for self confor- mal measures.J

    Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Logarithmic Fourier decay for self confor- mal measures.J. Lond. Math. Soc. (2), 106(2):1628–1661, 2022

    2022

  3. [3]

    Polynomial fourier decay and a cocycle version of Dolgopyat’s method for self conformal measures.arXiv preprint arXiv:2306.01275, 2023

    Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Polynomial fourier decay and a cocycle version of Dolgopyat’s method for self conformal measures.arXiv preprint arXiv:2306.01275, 2023

    arXiv 2023

  4. [4]

    Pointwise normality and Fourier decay for self-conformal measures.Adv

    Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Pointwise normality and Fourier decay for self-conformal measures.Adv. Math., 393:Paper No. 108096, 72, 2021

    2021

  5. [5]

    Spectral gaps and Fourier decay for self- conformal measures on the plane.Transactions of the American Mathematical Society, 379(6):3953–3991, 2026

    Amir Algom, Federico Rodriguez Hertz, and Zhiren Wang. Spectral gaps and Fourier decay for self- conformal measures on the plane.Transactions of the American Mathematical Society, 379(6):3953–3991, 2026

    2026

  6. [6]

    On the dimension of orthogonal projections of self-similar measures

    Amir Algom and Pablo Shmerkin. On the dimension of orthogonal projections of self-similar measures. J. Lond. Math. Soc. (2), 112(1):Paper No. e70245, 20, 2025. SMOOTH PROJECTIONS OF SELF-SIMILAR MEASURES 51

    2025

  7. [7]

    Disintegration results for fractal measures and applications to Diophantine approxima- tion.Ergodic Theory Dynam

    Simon Baker. Disintegration results for fractal measures and applications to Diophantine approxima- tion.Ergodic Theory Dynam. Systems, 46(4):885–902, 2026

    2026

  8. [8]

    Polynomial Fourier decay for fractal measures and their pushforwards

    Simon Baker and Amlan Banaji. Polynomial Fourier decay for fractal measures and their pushforwards. Mathematische Annalen, 392(1):209–261, 2025

    2025

Show all 62 references
  1. [9]

    Fourier decay froml 2-flattening.arXiv preprint arXiv:2407.16699, 2024

    Simon Baker, Osama Khalil, and Tuomas Sahlsten. Fourier decay froml 2-flattening.arXiv preprint arXiv:2407.16699, 2024

    2024 arXiv

  2. [10]

    Spectral gaps and Fourier dimension for self-conformal sets with overlaps.arXiv preprint arXiv:2306.01389, 2023

    Simon Baker and Tuomas Sahlsten. Spectral gaps and Fourier dimension for self-conformal sets with overlaps.arXiv preprint arXiv:2306.01389, 2023

    2023 arXiv

  3. [11]

    Fourier transform of nonlinear images of self-similar measures: quantitative aspects.Peking Mathematical Journal, 2026

    Amlan Banaji and Han Yu. Fourier transform of nonlinear images of self-similar measures: quantitative aspects.Peking Mathematical Journal, 2026. To appear

    2026

  4. [12]

    Projections of self-affine sets onto lines.arXiv preprint arXiv:2607.14740, 2026

    Balázs Bárány, Antti Käenmäki, and István Kolossváry. Projections of self-affine sets onto lines.arXiv preprint arXiv:2607.14740, 2026

    2026 arXiv

  5. [13]

    Scaling limits of self-conformal mea- sures.arXiv preprint arXiv:2308.11399, 2023

    Balázs Bárány, Antti Käenmäki, Aleksi Pyörälä, and Meng Wu. Scaling limits of self-conformal mea- sures.arXiv preprint arXiv:2308.11399, 2023

    2023 arXiv

  6. [14]

    A spectral gap theorem in simple Lie groups.Invent

    Yves Benoist and Nicolas de Saxcé. A spectral gap theorem in simple Lie groups.Invent. Math., 205(2):337–361, 2016

    2016

  7. [15]

    Bourgain and A

    J. Bourgain and A. Gamburd. A spectral gap theorem inSUpdq.J. Eur. Math. Soc. (JEMS), 14(5):1455– 1511, 2012

    2012

  8. [16]

    On the spectral gap for finitely-generated subgroups ofSUp2q.In- vent

    Jean Bourgain and Alex Gamburd. On the spectral gap for finitely-generated subgroups ofSUp2q.In- vent. Math., 171(1):83–121, 2008

    2008

  9. [17]

    Local spectral gap in the group of euclidean isometries.International Mathematics Research Notices, 2020(2):466–486, 2020

    Rémi Boutonnet and Adrian Ioana. Local spectral gap in the group of euclidean isometries.International Mathematics Research Notices, 2020(2):466–486, 2020

    2020

  10. [18]

    Projections of Gibbs measures on self-conformal sets.Nonlinearity, 32(2):603–621, 2019

    Catherine Bruce and Xiong Jin. Projections of Gibbs measures on self-conformal sets.Nonlinearity, 32(2):603–621, 2019

    2019

  11. [19]

    Furstenberg sumset conjecture and Mandelbrot percolations.arXiv preprint arXiv:2211.16410, 2022

    Catherine Bruce and Xiong Jin. Furstenberg sumset conjecture and Mandelbrot percolations.arXiv preprint arXiv:2211.16410, 2022

    2022 arXiv

  12. [20]

    Daniel Mauldin

    Robert Cawley and R. Daniel Mauldin. Multifractal decompositions of Moran fractals.Adv. Math., 92(2):196–236, 1992

    1992

  13. [21]

    L. Clozel. Automorphic forms and the distribution of points on odd-dimensional spheres.Israel J. Math., 132:175–187, 2002

    2002

  14. [22]

    Dynamical self-similarity,L q -dimensions and Furstenberg slicing inR d.arXiv preprint arXiv:2409.04608, 2024

    Emilio Corso and Pablo Shmerkin. Dynamical self-similarity,L q -dimensions and Furstenberg slicing inR d.arXiv preprint arXiv:2409.04608, 2024

    2024 arXiv

  15. [23]

    Weighted restriction estimates and application to Falconer distance set problem.American Journal of Mathematics, 143(1):175–211, 2021

    Xiumin Du, Larry Guth, Yumeng Ou, Hong Wang, Bobby Wilson, and Ruixiang Zhang. Weighted restriction estimates and application to Falconer distance set problem.American Journal of Mathematics, 143(1):175–211, 2021

    2021

  16. [24]

    K. J. Falconer. Generalized dimensions of measures on self-affine sets.Nonlinearity, 12(4):877–891, 1999

    1999

  17. [25]

    Sixty years of fractal projections

    Kenneth Falconer, Jonathan Fraser, and Xiong Jin. Sixty years of fractal projections. InFractal geometry and stochastics V, pages 3–25. Springer, 2015

    2015

  18. [26]

    Seventy years of fractal projections.arXiv preprint arXiv:2602.22002, 2026

    Kenneth J Falconer. Seventy years of fractal projections.arXiv preprint arXiv:2602.22002, 2026

    2026

  19. [27]

    Falconer and Xiong Jin

    Kenneth J. Falconer and Xiong Jin. Exact dimensionality and projections of random self-similar mea- sures and sets.J. Lond. Math. Soc. (2), 90(2):388–412, 2014

    2014

  20. [28]

    Typical self-affine sets with non-empty interior.Asian Journal of Mathemat- ics, 27(5):621–638, 2024

    De-Jun Feng and Zhou Feng. Typical self-affine sets with non-empty interior.Asian Journal of Mathemat- ics, 27(5):621–638, 2024

    2024

  21. [29]

    Dimension theory of iterated function systems.Comm

    De-Jun Feng and Huyi Hu. Dimension theory of iterated function systems.Comm. Pure Appl. Math., 62(11):1435–1500, 2009

    2009

  22. [30]

    Dimensions of orthogonal projections of typical self-affine sets and mea- sures.arXiv preprint arXiv:2502.04000, 2025

    De-Jun Feng and Yu-Hao Xie. Dimensions of orthogonal projections of typical self-affine sets and mea- sures.arXiv preprint arXiv:2502.04000, 2025

    2025

  23. [31]

    On self-similar sets with overlaps and inverse theorems for entropy.Ann

    Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy.Ann. of Math. (2), 180(2):773–822, 2014

    2014

  24. [32]

    On self-similar sets with overlaps and inverse theorems for entropy inR d.Memoirs of the American Mathematical Society, 265(1287), 2020

    Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy inR d.Memoirs of the American Mathematical Society, 265(1287), 2020

    2020

  25. [33]

    Local entropy averages and projections of fractal measures

    Michael Hochman and Pablo Shmerkin. Local entropy averages and projections of fractal measures. Ann. of Math. (2), 175(3):1001–1059, 2012

    2012

  26. [34]

    Dimension of ergodic measures projected onto self-similar sets with overlap.Proceedings of the London Mathematical Society, 122(2):191–206, 2021

    Thomas Jordan and Ariel Rapaport. Dimension of ergodic measures projected onto self-similar sets with overlap.Proceedings of the London Mathematical Society, 122(2):191–206, 2021. 52 AMIR ALGOM, FEDERICO RODRIGUEZ HERTZ, AND ZHIREN WANG

    2021

  27. [35]

    On absolute continuity of inhomogeneous and contracting on average self-similar measures.arXiv preprint arXiv:2409.18936, 2024

    Samuel Kittle and Constantin Kogler. On absolute continuity of inhomogeneous and contracting on average self-similar measures.arXiv preprint arXiv:2409.18936, 2024

    2024

  28. [36]

    Elon Lindenstrauss and Péter P . Varjú. Random walks in the group of Euclidean isometries and self- similar measures.Duke Math. J., 165(6):1061–1127, 2016

    2016

  29. [37]

    Lubotzky, R

    A. Lubotzky, R. Phillips, and P . Sarnak. Hecke operators and distributing points on the sphere. I. vol- ume 39, pages S149–S186. 1986. Frontiers of the mathematical sciences: 1985 (New York, 1985)

    1986

  30. [38]

    Lubotzky, R

    A. Lubotzky, R. Phillips, and P . Sarnak. Hecke operators and distributing points onS 2. II.Comm. Pure Appl. Math., 40(4):401–420, 1987

    1987

  31. [39]

    Spherical averages of fourier transforms of measures with finite energy; dimensions of intersections and distance sets.Mathematika, 34(2):207–228, 1987

    Pertti Mattila. Spherical averages of fourier transforms of measures with finite energy; dimensions of intersections and distance sets.Mathematika, 34(2):207–228, 1987

    1987

  32. [40]

    Cambridge University Press, Cambridge, 1995

    Pertti Mattila.Geometry of sets and measures in Euclidean spaces, volume 44 ofCambridge Studies in Ad- vanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability

    1995

  33. [41]

    Cambridge University Press, Cambridge, 2015

    Pertti Mattila.Fourier analysis and Hausdorff dimension, volume 150 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2015

    2015

  34. [42]

    Morris and Ça˘ grı Sert

    Ian D. Morris and Ça˘ grı Sert. Projections of self-affine fractals.Inventiones Mathematicae, 2026. To appear

    2026

  35. [43]

    Convolutions of Cantor measures without resonance

    Fedor Nazarov, Yuval Peres, and Pablo Shmerkin. Convolutions of Cantor measures without resonance. Israel J. Math., 187:93–116, 2012

    2012

  36. [44]

    The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group

    Hee Oh. The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group. volume 149, pages 301–316. 2005. Probability in mathematics

    2005

  37. [45]

    Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions.Duke Math

    Yuval Peres and Wilhelm Schlag. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions.Duke Math. J., 102(2):193–251, 2000

    2000

  38. [46]

    Resonance between Cantor sets.Ergodic Theory Dynam

    Yuval Peres and Pablo Shmerkin. Resonance between Cantor sets.Ergodic Theory Dynam. Systems, 29(1):201–221, 2009

    2009

  39. [47]

    Existence ofL q dimensions and entropy dimension for self-conformal measures.Indiana Univ

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

    2000

  40. [48]

    The dimension of projections of planar diagonal self-affine measures.Ann

    Aleksi Pyörälä. The dimension of projections of planar diagonal self-affine measures.Ann. Fenn. Math., 50(1):59–78, 2025

    2025

  41. [49]

    A self-similar measure with dense rotations, singular projections and discrete slices

    Ariel Rapaport. A self-similar measure with dense rotations, singular projections and discrete slices. Adv. Math., 321:529–546, 2017

    2017

  42. [50]

    On self-similar measures with absolutely continuous projections and dimension con- servation in each direction.Ergodic Theory Dynam

    Ariel Rapaport. On self-similar measures with absolutely continuous projections and dimension con- servation in each direction.Ergodic Theory Dynam. Systems, 40(12):3438–3456, 2020

    2020

  43. [51]

    Fourier transforms and iterated function systems

    Tuomas Sahlsten. Fourier transforms and iterated function systems. InRecent developments in fractals and related fields, Trends Math., pages 297–346. Birkhäuser/Springer, Cham, [2025] ©2025

    2025

  44. [52]

    Springer International Publishing, Cham, 2015

    Pablo Shmerkin.Projections of Self-Similar and Related Fractals: A Survey of Recent Developments, pages 53–74. Springer International Publishing, Cham, 2015

    2015

  45. [53]

    On Furstenberg’s intersection conjecture, self-similar measures, and theL q norms of convolutions.Ann

    Pablo Shmerkin. On Furstenberg’s intersection conjecture, self-similar measures, and theL q norms of convolutions.Ann. of Math. (2), 189(2):319–391, 2019

    2019

  46. [54]

    Absolute continuity of self-similar measures, their projections and convolutions.Trans

    Pablo Shmerkin and Boris Solomyak. Absolute continuity of self-similar measures, their projections and convolutions.Trans. Amer. Math. Soc., 368(7):5125–5151, 2016

    2016

  47. [55]

    Estimates of spherical averages of Fourier transforms and dimensions of sets.Mathematika, 40(2):322–330, 1993

    Per Sjölin. Estimates of spherical averages of Fourier transforms and dimensions of sets.Mathematika, 40(2):322–330, 1993

    1993

  48. [56]

    Notes on the transversality method for iterated function systems—a survey.Mathe- matical and Computational Applications, 28(3):65, 2023

    Boris Solomyak. Notes on the transversality method for iterated function systems—a survey.Mathe- matical and Computational Applications, 28(3):65, 2023

    2023

  49. [57]

    Fourier decay and absolute continuity for typical homogeneous self-similar measures inR d fordě3.arXiv preprint arXiv:2508.14698, 2025

    Boris Solomyak. Fourier decay and absolute continuity for typical homogeneous self-similar measures inR d fordě3.arXiv preprint arXiv:2508.14698, 2025

    2025 arXiv

  50. [58]

    Absolute continuity of self-similar measures on the plane.Indiana University Mathematics Journal, 74(4):1023–1097, 2025

    Boris Solomyak and Adam ´Spiewak. Absolute continuity of self-similar measures on the plane.Indiana University Mathematics Journal, 74(4):1023–1097, 2025

    2025

  51. [59]

    Taylor.Partial differential equations, volume 23 ofTexts in Applied Mathematics

    Michael E. Taylor.Partial differential equations, volume 23 ofTexts in Applied Mathematics. Springer- Verlag, New York, 1996. Basic theory

    1996

  52. [60]

    Birkhäuser, Basel, 1983

    Hans Triebel.Theory of Function Spaces, volume 78 ofMonographs in Mathematics. Birkhäuser, Basel, 1983

    1983

  53. [61]

    Random walks in compact groups.Documenta Mathematica, 18:1137–1175, 2013

    Péter Pál Varjú. Random walks in compact groups.Documenta Mathematica, 18:1137–1175, 2013

    2013

  54. [62]

    Projection theorems with countably many exceptions and applications to the exact overlaps conjecture.arXiv preprint arXiv:2503.21923, 2025

    Meng Wu. Projection theorems with countably many exceptions and applications to the exact overlaps conjecture.arXiv preprint arXiv:2503.21923, 2025. SMOOTH PROJECTIONS OF SELF-SIMILAR MEASURES 53 DEPARTMENT OFMATHEMATICS, UNIVERSITY OFHAIFA ATORANIM, TIVON36006, ISRAEL Email a...

    2025

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