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Distinct Distances on Pfaffian Curves

Abhiram Natarajan, Adam Sheffer

TL;DR

This work extends the distinct-distances bound to Pfaffian curves, placing the problem in the tame, o-minimal Pfaffian setting and enabling bounds that mirror the algebraic case. The authors develop a distance-energy framework and an incidence-based approach, adapting projections and symmetry arguments to Pfaffian curves and leveraging the Solymosi–Zahl proximity technique. The main result establishes $D(\mathcal{P}_1,\mathcal{P}_2) = \Omega_{\alpha,\beta,r}(\min\{m^{3/4}n^{3/4}, m^2, n^2\})$ for two Pfaffian curves with the stated exclusions, and a corollary yields $D(\mathcal{P}) = \Omega_{\alpha,\beta,r}(n^{3/2})$ for a single curve. The work advances discrete geometry by integrating Pfaffian/topological tameness tools into incidence-bound techniques, suggesting pathways to broader generalizations within tame geometry.

Abstract

We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains $Ω(\min\{m^{3/4}n^{3/4},m^2,n^2\})$, as obtained from the proximity technique of Solymosi and Zahl.

Distinct Distances on Pfaffian Curves

TL;DR

This work extends the distinct-distances bound to Pfaffian curves, placing the problem in the tame, o-minimal Pfaffian setting and enabling bounds that mirror the algebraic case. The authors develop a distance-energy framework and an incidence-based approach, adapting projections and symmetry arguments to Pfaffian curves and leveraging the Solymosi–Zahl proximity technique. The main result establishes for two Pfaffian curves with the stated exclusions, and a corollary yields for a single curve. The work advances discrete geometry by integrating Pfaffian/topological tameness tools into incidence-bound techniques, suggesting pathways to broader generalizations within tame geometry.

Abstract

We generalize Pach and de Zeeuw's bound for distinct distances between points on two curves, from algebraic curves to Pfaffian curves. Pfaffian curves include those that can be defined by any combination of elementary functions, including exponential and logarithmic functions, rational and irrational powers, trigonometric functions and their inverses, integration, and more. The bound remains , as obtained from the proximity technique of Solymosi and Zahl.

Paper Structure

This paper contains 6 sections, 17 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Proof of Theorem \ref{['th:twocurves']}
  4. Proof of Lemma \ref{['le:CijInt']}
  5. Symmetries of Pfaffian sets
  6. Proof of Lemma \ref{['le:CijInt']}-\ref{['cij:no-three-infinite']}

Key Result

theorem 1

Let $\gamma_1,\gamma_2\subset\mathbb R^2$ be algebraic curves of degree at most $k$. Let $\mathcal{P}_1$ be a set of $m$ points on $\gamma_1$ and let $\mathcal{P}_2$ be a set of $n$ points on $\gamma_2$. If $\gamma_1$ and $\gamma_2$ are neither parallel lines, orthogonal lines, nor concentric circle

Figures (1)

  • Figure 1: A log-circles configuration

Theorems & Definitions (28)

  • theorem 1
  • theorem 2: store=thtwocurves
  • corollary 1
  • corollary 2
  • theorem 3
  • theorem 4
  • proof
  • theorem 5
  • corollary 3
  • proof
  • ...and 18 more