Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Projection Theorems in the Presence of Expansions

K. W. Ohm, Z. Lin

TL;DR

This work proves a restricted projection theorem for a specific one-dimensional family of projections $\\pi_{t,r}$ from $\\mathbb{R}^3$ to $\\mathbb{R}^2$, deriving a sharp decay bound on the $\\delta$-neighborhood multiplicity of projected points for most parameters $r$ outside a small exceptional set, with the bound depending on the original measure dimension $\\alpha$. The authors first tackle the $3$-dimensional case using OV-Planes-inspired arguments, and then extend the results to higher dimensions by leveraging the decoupling results of Gan–Guo–Wang (GGW). They also formulate a dimension-improvement result in irreducible $\\mathrm{SL}_2(\\mathbb{R})$-representations, connecting the projection problem to representation-theoretic expansions via the actions of $a_t$ and $u_r$. The combination of elementary geometric arguments with modern decoupling tools yields quantitative, nearly-optimal bounds with explicit exceptional-parameter control, providing applications to homogeneous dynamics and related incidence problems. The results illustrate how tailored projection families interact with distributional hypotheses to yield robust dimension preservation or improvement statements.

Abstract

We prove a restricted projection theorem for a certain one dimensional family of projections from $\mathbb R^n$ to $\mathbb R^k$. The family we consider here arises naturally in the study of quantitative equidistribution problems in homogeneous dynamics.

Projection Theorems in the Presence of Expansions

TL;DR

This work proves a restricted projection theorem for a specific one-dimensional family of projections from to , deriving a sharp decay bound on the -neighborhood multiplicity of projected points for most parameters outside a small exceptional set, with the bound depending on the original measure dimension . The authors first tackle the -dimensional case using OV-Planes-inspired arguments, and then extend the results to higher dimensions by leveraging the decoupling results of Gan–Guo–Wang (GGW). They also formulate a dimension-improvement result in irreducible -representations, connecting the projection problem to representation-theoretic expansions via the actions of and . The combination of elementary geometric arguments with modern decoupling tools yields quantitative, nearly-optimal bounds with explicit exceptional-parameter control, providing applications to homogeneous dynamics and related incidence problems. The results illustrate how tailored projection families interact with distributional hypotheses to yield robust dimension preservation or improvement statements.

Abstract

We prove a restricted projection theorem for a certain one dimensional family of projections from to . The family we consider here arises naturally in the study of quantitative equidistribution problems in homogeneous dynamics.
Paper Structure (3 sections, 6 theorems, 96 equations, 1 figure)

This paper contains 3 sections, 6 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm: main finitary']}
  3. Proof of Theorem \ref{['thm:Improving Dimension SL2 irrrep']}

Key Result

Theorem 1.1

Let $1\leq \alpha\leq 3/2$, and let $0<\delta_0\leq1$. Let $F\subset B_{\mathbb{R}^3}(0,1)$ be a finite set and let $\mu$ denote the uniform measure on $F$. Assume that where $C_0\geq 1$. Let $0<\varepsilon<\alpha/100$. For every $\delta\geq e^t\delta_0$, there is a subset there exists $\mathcal{E} = \mathcal{E}_{{t}, \delta} \subset [0,1]$ with $|\mathcal{E}|\ll_\varepsilon \delta^{\star\varepsi

Figures (1)

  • Figure 1: Graph of the piecewise function $\varpi(\cdot)$

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Remark
  • Remark
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 5 more