Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Turán and Delsarte problems and their duals

Mihail N. Kolountzakis, Nir Lev, Máté Matolcsi

TL;DR

The work develops a unified Euclidean framework for the Turán and Delsarte extremal problems and their duals, proving weak and strong linear duality, and establishing the existence of extremizers along with tiling-type relations between primal extremizers and dual measures. It then applies these results to the Delsarte packing bound, linking density limits to the essential difference set and to tiling/spectrality concepts in convex bodies. The findings show that the Delsarte bound strictly improves the trivial volume bound for any convex body that does not tile, and they unify finite-group intuitions with continuous theory, offering a path toward resolving questions about Turán domains and spectral-tiling correspondences in Euclidean space. Overall, the paper provides a robust toolkit for assessing packing densities and tiling properties through dual optimization problems in analysis and convex geometry.

Abstract

We study two optimization problems for positive definite functions on Euclidean space with restrictions on their support and sign: the Turan problem and the Delsarte problem. These problems have been studied also for their connections to geometric problems of tiling and packing. In the finite group setting the weak and strong linear duality for these problems are automatic. We prove these properties in the continuous setting. We also show the existence of extremizers for these problems and their duals, and establish tiling-type relations between the extremal functions for each problem and the extremal measures or distributions for the dual problem. We then apply the results to convex bodies, and prove that the Delsarte packing bound is strictly better than the trivial volume packing bound for every convex body that does not tile the space.

The Turán and Delsarte problems and their duals

TL;DR

The work develops a unified Euclidean framework for the Turán and Delsarte extremal problems and their duals, proving weak and strong linear duality, and establishing the existence of extremizers along with tiling-type relations between primal extremizers and dual measures. It then applies these results to the Delsarte packing bound, linking density limits to the essential difference set and to tiling/spectrality concepts in convex bodies. The findings show that the Delsarte bound strictly improves the trivial volume bound for any convex body that does not tile, and they unify finite-group intuitions with continuous theory, offering a path toward resolving questions about Turán domains and spectral-tiling correspondences in Euclidean space. Overall, the paper provides a robust toolkit for assessing packing densities and tiling properties through dual optimization problems in analysis and convex geometry.

Abstract

We study two optimization problems for positive definite functions on Euclidean space with restrictions on their support and sign: the Turan problem and the Delsarte problem. These problems have been studied also for their connections to geometric problems of tiling and packing. In the finite group setting the weak and strong linear duality for these problems are automatic. We prove these properties in the continuous setting. We also show the existence of extremizers for these problems and their duals, and establish tiling-type relations between the extremal functions for each problem and the extremal measures or distributions for the dual problem. We then apply the results to convex bodies, and prove that the Delsarte packing bound is strictly better than the trivial volume packing bound for every convex body that does not tile the space.

Paper Structure

This paper contains 33 sections, 26 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The Turán and Delsarte problems in finite groups
  3. The Turán problem
  4. The Delsarte problem
  5. Difference sets
  6. Euclidean setting preliminaries
  7. The Turán problem and its dual
  8. Admissible domains for the Turán problem
  9. The Turán constant
  10. Difference sets and convex domains
  11. The dual Turán constant
  12. Weak linear duality in the Turán problem
  13. Strong linear duality in the Turán problem
  14. Remark
  15. Existence of extremizers for the Turán problem and its dual
  16. ...and 18 more sections

Key Result

Theorem 2.1

$T(U)T'(U)=1$ holds for every $U \subset G$ with $0 \in U = - U$.

Theorems & Definitions (50)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Lemma 3.1: see KL21
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3
  • ...and 40 more