[Paper Review] Sets of large dimension not containing polynomial configurations

András Máthé|Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)|Jan 2, 2012

Mathematical Dynamics and Fractals6 references4 citations

TL;DR

This paper introduces a general method to construct compact sets in $ℝ^n$ of Hausdorff dimension $n/d$ that avoid finite point configurations defined by countably many multivariate polynomials with rational coefficients and maximum degree $d$. The key contribution is a constructive framework that yields sets avoiding specific geometric configurations—such as angles, distances, or collinear points—while achieving optimal dimension bounds, including sets of dimension $n/2$ avoiding the right angle $\pi/2$, and sets of dimension $n/8$ avoiding any given angle.

ABSTRACT

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree $d$, we construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/d$ which does not contain finite point configurations corresponding to the zero sets of the given polynomials. Given a set $E\subset \R^n$, we study the angles determined by three points of $E$. The main result implies the existence of a compact set in $\R^n$ of Hausdorff dimension $n/2$ which does not contain the angle $π/2$. (This is known to be sharp if $n$ is even.) We show that there is a compact set of Hausdorff dimension $n/8$ which does not contain an angle in any given countable set. We also construct a compact set $E\subset \R^n$ of Hausdorff dimension $n/6$ for which the set of angles determined by $E$ is Lebesgue null. In the other direction, we present a result that every set of sufficiently large dimension contains an angle $ε$ close to any given angle.

Motivation & Objective

• To develop a general method for constructing compact sets in $ℝ^n$ of controlled Hausdorff dimension that avoid finite point configurations defined by multivariate polynomials with rational coefficients.
• To address open problems on the minimal dimension required for a set to contain specific angles or distance configurations.
• To strengthen Falconer's distance set conjecture by constructing sets of dimension $n/2$ with Lebesgue null distance sets and additional exclusions (e.g., no rational distances, no collinear points).
• To provide sharp dimension thresholds for avoiding specific angles, including $\pi/2$, $\pi/3$, and $2\pi/3$, and to show that such exclusions are possible even for irrational angles.

Proposed method

• The method constructs a decreasing sequence of compact sets $F^k$ via a recursive, measure-theoretic construction using dyadic balls and a mass distribution principle.
• It uses a parameterized family of sets $E_j''$ and a nested construction with parameters $b_j$, $h_j$, and $r_j$ to control the density and dimension of the limit set.
• The construction ensures that for any given polynomial $P_j$ of degree $d$, no $m_j$-tuple of points in the final set $F$ satisfies $P_j = 0$, by ensuring that such configurations are excluded at each stage.
• A key step involves bounding the number of points in a ball of radius $r$ intersecting the current set, using estimates based on $h_j$, $b_j$, and $r$, leading to a dimension bound via the mass distribution principle.
• The method applies to polynomials in $3n$ variables defining angles (e.g., $\langle y-x,z-x\rangle = 0$ for $\pi/2$), and extends to rational cosine values and even irrational angles via algebraic approximation.
• The final set $F$ is shown to have Hausdorff dimension at least $n/d$ by proving that the associated probability measure $\mu$ satisfies $\mu(B(x,r)) \leq C r^{n/d} (\log r^{-1})^{n+1}$, implying the dimension lower bound.

Experimental results

Research questions

• RQ1 What is the minimal Hausdorff dimension of a compact set in $\mathbb{R}^n$ that avoids a given angle $\alpha$?
• RQ2 Can a set of dimension $n/2$ avoid the right angle $\pi/2$, and is this dimension optimal?
• RQ3 Is it possible to construct a set of dimension $n/8$ that avoids any given angle $\alpha \in [0,\pi]$?
• RQ4 Can a set of dimension $n/6$ be constructed such that its set of angles has Lebesgue measure zero?
• RQ5 What is the largest possible dimension of a Borel set in $\mathbb{R}^n$ that avoids all rational distances and collinear points?

Key findings

• There exists a compact set $E \subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ that does not contain the angle $\pi/2$, and this bound is sharp when $n$ is even.
• For any countable set of angles, there exists a compact set of Hausdorff dimension $n/8$ that avoids all angles in the set.
• There exists a compact set $E \subset \mathbb{R}^n$ of Hausdorff dimension $n/6$ such that the set of angles determined by $E$ has Lebesgue measure zero.
• A compact set $E \subset \mathbb{R}^n$ of dimension $n/2$ can be constructed such that $D(E)$ is Lebesgue null, contains no rational distances, no collinear points, and every distance and direction is realized at most once.
• The method applies to exclude configurations defined by any countably many polynomials with rational coefficients and maximum degree $d$, yielding sets of dimension $n/d$ avoiding all such configurations.
• The construction shows that for any given angle $\alpha$, a compact set of dimension $n/8$ exists that avoids $\alpha$, even when $\alpha$ is irrational, by leveraging rational approximations of cosine values.