---
title: The hybrid spectral test
---

\[section\] \[Thm\][Proposition]{} \[Thm\][Lemma]{} \[Thm\][Conjecture]{} \[Thm\][Corollary]{} \[Thm\][Example]{} \[Thm\][Definition]{} \[Thm\][Research Problem]{} \[Thm\][Remark]{}

Introduction {#s:introduction}
============

This paper is about certain mathematical concepts to analyze the uniform distribution behaviour of sequences on the $s$-dimensional torus $[0,1)^s$. We discuss qualitative aspects, by which we understand the study of properties of an infinite sequence that induce its uniform distribution in $[0,1)^s$, and also quantitative aspects, i.e. the question how to measure the uniform distribution of a finite or infinite sequence on the $s$-torus.

A general setting to generate a (finite or infinite) sequence $\omega=(x_n)_{n\ge 0}$ in some output space ${\ensuremath{\mathcal{O}}}$, like ${\ensuremath{\mathcal{O}}}=[0,1)^s$, is the following. We consider a nonempty set ${\ensuremath{\mathcal{S}}}$, the so-called state space, and a map $T:{\ensuremath{\mathcal{S}}}\rightarrow {\ensuremath{\mathcal{S}}}$, the state update transformation. We employ $T$ to generate a (finite or infinite) sequence of states $(s_n)_{n\ge 0}$ in ${\ensuremath{\mathcal{S}}}$. This sequence is then mapped to a sequence $\omega=(x_n)_{n\ge 0}$ in ${\ensuremath{\mathcal{O}}}$ by an output map $\varphi: {\ensuremath{\mathcal{S}}}\rightarrow {\ensuremath{\mathcal{O}}}$, by letting $x_n=\varphi(s_n)$, $n\ge 0$.

In numerous applications of this concept, the sequence $(s_n)_{n\ge 0}$ is constructed by iterating $T$. We start with some initial state $s_0$ and put $s_n=T^n(s_0)$, $n\ge 0$. Illustrative examples are linear or inversive congruential 