Completeness of sparse, almost integer and finite local complexity sequences of translates in $L^p(\mathbb{R})$

TL;DR

The paper advances the understanding of completeness by translates in $L^p(\mathbb{R})$ for the challenging range $1<p\le2$, showing that $p$-generating sets need not be lacunary or arithmetic. It proves three key results: (i) the existence of very sparse $p$-generating sequences with $\lambda_{n+1}/\lambda_n>1+\varepsilon_n$ and a nonnegative generator in $\bigcap_{p>1} L^p(\mathbb{R})$; (ii) that every almost-integer sequence $\lambda_n=n+\alpha_n$, with $\alpha_n\to0$ and $\alpha_n\neq0$, is $p$-generating for all $p>1$ via a nonnegative generator; and (iii) the existence of finite local complexity generating sequences with only two gap values, e.g., $\lambda_{n+1}-\lambda_n\in\{a,b\}$ with $a,b>0$ linearly independent over $\mathbb{Q}$, using a Schwartz generator. The methods center on Landau-system constructions, $A^p$-space completeness of weighted exponentials, and intricate perturbation arguments to realize generators with nonnegativity and smoothness properties. Collectively, the results reveal that both extreme sparsity and rigid gap structures can still yield complete translate systems, enriching the landscape between Beurling-type density results and lacunarity constraints.

Abstract

A real sequence $Λ= \{λ_n\}_{n=1}^\infty$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-λ_n)\}_{n=1}^\infty$ span the space $L^p(\mathbb{R})$. While the $p$-generating sets were completely characterized for $p=1$ and $p>2$, the case $1 < p \le 2$ remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In the present paper, (i) We show that a $p$-generating set $Λ$ of positive real numbers can be very sparse, namely, the ratios $λ_{n+1} / λ_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every "almost integer" sequence $Λ$, i.e. satisfying $λ_n = n + α_n$, $0 \neq α_n \to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $Λ$ such that the successive differences $λ_{n+1} - λ_n$ attain only two different positive values. The constructions are, in a sense, extreme: it is well known that $Λ$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.

Theorems & Definitions (28)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Definition 4.1
• Definition 4.2
• Proposition 4.3
• proof
• Definition 4.4
• Theorem 4.5