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Uniformly perfect measures on strictly convex planar graphs are $L^{2}$-flattening

Amir Algom, Tuomas Orponen

TL;DR

The paper proves an $L^{2}$-flattening phenomenon for uniformly perfect measures supported on strictly convex planar $C^{2}$ graphs, showing that for any such measure $\sigma$ there exists $p\ge 1$ with $\|\hat{\sigma}\|_{L^{p}(B(R))}^{p} \lesssim R^{\varepsilon}$ for all $R\ge 1$ and any $\varepsilon>0$. The argument blends one-dimensional inverse theorems (Rossi–Shmerkin and Shmerkin) with a geometric decompositional approach to the planar graph, exploiting curvature to force sumset growth and, consequently, decay in the Fourier side after mollification. Key steps include establishing $L^{2}$-flattening for convolutions with uniformly perfect measures, a refined Frostman-type control, and an iterative amplification of flattening through repeated convolutions. The results extend prior line-based flattening to curved curves in the plane, with potential implications for Fourier decay problems for measures on curves and surfaces, and complement existing affine non-concentration theories in higher dimensions.

Abstract

Uniformly perfect measures are a common generalisation of Ahlfors regular measures, self-conformal measures on the line, and their push-forwards under sufficiently regular maps. We show that every uniformly perfect measure $σ$ on a strictly convex planar $C^{2}$-graph is $L^{2}$-flattening. That is, for every $ε>0$, there exists $p = p(ε,σ) \geq 1$ such that $$\|\hatσ\|_{L^{p}(B(R))}^{p} \lesssim_{ε,σ} R^ε, \qquad R \geq 1.$$

Uniformly perfect measures on strictly convex planar graphs are $L^{2}$-flattening

TL;DR

The paper proves an -flattening phenomenon for uniformly perfect measures supported on strictly convex planar graphs, showing that for any such measure there exists with for all and any . The argument blends one-dimensional inverse theorems (Rossi–Shmerkin and Shmerkin) with a geometric decompositional approach to the planar graph, exploiting curvature to force sumset growth and, consequently, decay in the Fourier side after mollification. Key steps include establishing -flattening for convolutions with uniformly perfect measures, a refined Frostman-type control, and an iterative amplification of flattening through repeated convolutions. The results extend prior line-based flattening to curved curves in the plane, with potential implications for Fourier decay problems for measures on curves and surfaces, and complement existing affine non-concentration theories in higher dimensions.

Abstract

Uniformly perfect measures are a common generalisation of Ahlfors regular measures, self-conformal measures on the line, and their push-forwards under sufficiently regular maps. We show that every uniformly perfect measure on a strictly convex planar -graph is -flattening. That is, for every , there exists such that

Paper Structure

This paper contains 19 sections, 19 theorems, 235 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous work
  3. Proof outline
  4. Acknowledgements
  5. Preliminaries
  6. Uniform sets
  7. Rossi and Shmerkin's theorem
  8. Proof of Proposition \ref{['RSProp']}
  9. Geometry of uniformly perfect measures
  10. Further auxiliary lemmas
  11. Main technical proposition
  12. Proof of Lemma \ref{['lemma1']}
  13. Choice of parameters and an assumption towards a contradiction
  14. Preliminary definitions and constructions
  15. Slices and projections of our chosen cube
  16. ...and 4 more sections

Key Result

Theorem 1.2

For every $D \geq 1$, $\mathfrak{d} > 0$, $\beta \in (0,1]$, and $\epsilon \in (0,1)$ there exists $p=p(D,\beta,\epsilon)\geq 1$ such that the following holds. Let $\sigma$ be a $(D,\beta)$-uniformly perfect probability measure with $\operatorname{spt} \sigma \subset \mathbb{P}$ and $\operatorname{d

Figures (1)

  • Figure 1: Left: the set $Y \subset \mathbb{P}$ and the $\Delta$-disc $B$. Right: the structure of $X$ under the hypothesis $|X \cap Y_{B}|_{\delta} \approx |X|_{\delta}$.

Theorems & Definitions (70)

  • Definition 1.1: $(D,\beta)$-uniformly perfect measure
  • Theorem 1.2
  • proof : Proof sketch
  • Definition 2.1: Uniform set
  • Proposition 2.2
  • Definition 2.3: $\delta$-measures and their $L^{2}$-norm
  • Definition 2.4: $(D,\beta,U)$-uniformly perfect measure
  • Remark 2.5
  • Proposition 2.6: Rossi-Shmerkin
  • Theorem 2.7: Shmerkin's inverse theorem
  • ...and 60 more