Spectral Envelopes - A Preliminary Report

TL;DR

This paper explores the spectral envelope of integer subsets, linking harmonic analysis with dynamical systems, and investigates implications for the Kadison-Singer problem, quasicrystals, and groups with Kazhdan's property T.

Contribution

It provides a partial characterization of spectral envelopes, relates them to dynamical properties, and extends the analysis to groups with Kazhdan's property T, offering new insights into longstanding problems.

Findings

Spectral envelopes are convex for minimal characteristic functions.

Numerical methods reveal properties of Morse-Thue sequences.

Results relate spectral properties to quasicrystals and group theory.

Abstract

The spectral envelope S(F) of a subset of integers is the set of probability measures on the circle group that are weak star limits of squared moduli of trigonometric polynomials with frequencies in F. Fourier transforms of these measures are positive and supported in F - F but the converse generally fails. The characteristic function chiF of F is a binary sequence whose orbit closure gives a symbolic dynamical system O(F). Analytic properties of S(F) are related to dynamical properties of chiF. The Riemann-Lebesque lemma implies that if chiF is minimal, then S(F) is convex and hence S(F) is the closure of the convex hull of its extreme points Se(F). In this paper we (i) review the relationship between these concepts and the special case of the still open 1959 Kadison-Singer problem called Feichtinger's conjecture for exponential functions, (ii) partially characterize of elements in Se(F), for minimal chiF, in terms of ergodic properties of (O(F),lambda) where lambda is a shift invariant probability measure whose existence in ensured by the 1937 Krylov-Bogoyubov theorem, (iii) refine previous numerical studies of the Morse-Thue minimal binary sequence by exploiting a new MATLAB algorithm for computing smallest eigenvalues of 4,000,000 x 4,000,000 matrices, (iv) describe recent results characterizing S(F) for certain Bohr sets F related to quasicrystals, (v) extend these concepts to general discrete groups including those with Kazhdan's T-property, such as SL(n,Z), n > 2, which can be characterized by several equivalent properties such as: any sequence of positive definite functions converging to 1 uniformly on compact subsets converges uniformly. This exotic property may be useful to construct a counterexample to the generalization of Feichtinger's conjecture and hence to provide a no answer to the question of Kadison and Singer whcih they themselves tended to suspect.