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Distinct Distances Between a Line and Strip

Sanjana Das, Adam Sheffer

TL;DR

This work studies distinct distances between a line-placed point set of size $m$ and a two-dimensional strip around another line containing $n$ points, under a spacing condition. It generalizes Purdy’s line-pair problem and leverages the Solymosi–Zahl proximity framework, combining a proximity-based ingredient with incidence bounds to produce a lower bound of $|\Delta(\mathcal{P}_1,\mathcal{P}_2)| = \Omega\big( \min\{ m^{14/15}n^{8/15-\varepsilon}, m^{3/4}n^{3/4}, m^2, n^2 \} \big)$ for nonparallel, nonorthogonal lines, with constants depending on $u$, $w$, and the angle. The paper extends the approach to nonlinear strips by allowing $\mathcal{P}_2$ to lie in a strip around a Lipschitz, $k$-nice curve $\{(f(y),y)\}$, under analogous spacing and monotonicity conditions, yielding the same type of bound. The core strategy combines a refined analysis of proximity-sensitive pair types (short, steep, shallow), a lower bound on a restricted distance energy $\mathsf{E}_t$, and an upper bound via the Sharir–Zahl incidence bound, followed by asymptotic optimization over the proximity parameter. This framework advances the understanding of how proximity constraints sharpen distinct-distance bounds and provides a pathway for nonlinear-strip configurations in incidence geometry.

Abstract

We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances between points on two lines or curves. Unlike these existing problems, this new variant only makes sense if the points satisfy an additional spacing condition. Our work can also be seen as an exploration of the proximity technique that was recently introduced by Solymosi and Zahl. This technique lies at the heart of our analysis.

Distinct Distances Between a Line and Strip

TL;DR

This work studies distinct distances between a line-placed point set of size and a two-dimensional strip around another line containing points, under a spacing condition. It generalizes Purdy’s line-pair problem and leverages the Solymosi–Zahl proximity framework, combining a proximity-based ingredient with incidence bounds to produce a lower bound of for nonparallel, nonorthogonal lines, with constants depending on , , and the angle. The paper extends the approach to nonlinear strips by allowing to lie in a strip around a Lipschitz, -nice curve , under analogous spacing and monotonicity conditions, yielding the same type of bound. The core strategy combines a refined analysis of proximity-sensitive pair types (short, steep, shallow), a lower bound on a restricted distance energy , and an upper bound via the Sharir–Zahl incidence bound, followed by asymptotic optimization over the proximity parameter. This framework advances the understanding of how proximity constraints sharpen distinct-distance bounds and provides a pathway for nonlinear-strip configurations in incidence geometry.

Abstract

We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances between points on two lines or curves. Unlike these existing problems, this new variant only makes sense if the points satisfy an additional spacing condition. Our work can also be seen as an exploration of the proximity technique that was recently introduced by Solymosi and Zahl. This technique lies at the heart of our analysis.

Paper Structure

This paper contains 13 sections, 11 theorems, 59 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Our problem and main result
  4. A statement for nonlinear strips
  5. Overview
  6. The main ingredient for proximity
  7. A preliminary observation
  8. Steep pairs
  9. Shallow pairs
  10. The proximity argument
  11. A lower bound
  12. An upper bound
  13. Final computations

Key Result

Theorem 1.1

Fix $u, w > 0$. Let $\ell_1$ and $\ell_2$ be two lines which are not parallel or orthogonal, and let $\mathcal{P}_1 \subseteq \ell_1$ and $\mathcal{P}_2 \subseteq \mathsf{Strip}_w(\ell_2)$ be $u$-spaced sets of points with $\left\lvert \mathcal{P}_1 \right\rvert = m$ and $\left\lvert \mathcal{P}_2 \ where the implicit constant depends on $\varepsilon$, $u$, $w$, and the angle between $\ell_1$ and

Figures (8)

  • Figure 1: A construction with $\Theta(n^2/\sqrt{\log n})$ distances.
  • Figure 2: We still require $\mathcal{P}_1$ to lie on $\ell_1$, but we only require $\mathcal{P}_2$ to lie on a strip around $\ell_2$.
  • Figure 3: An illustration of why some spacing condition is necessary for the problem to be meaningful.
  • Figure 4: An illustration of why $f(y) = sy$ is $2$-nice.
  • Figure 5: For example, Theorem \ref{['thm:gen-curves']} applies if we take $\mathcal{P}_2$ to lie in a strip around the curve $y^2 = x^3 + 1$.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Claim 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 10 more