Completeness of sparse, almost integer and finite local complexity sequences of translates in L^p(mathbb{R})
Pith reviewed 2026-05-23 03:16 UTC · model grok-4.3
The pith
For 1 < p ≤ 2, p-generating sets in L^p(R) can have ratios approaching 1 arbitrarily slowly, consist of almost-integers, or use only two successive differences.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
A real sequence Λ is p-generating if there exists g such that the translates {g(x-λ_n)} span L^p(R). For 1<p≤2 the paper constructs such Λ that are arbitrarily sparse in the sense that λ_{n+1}/λ_n →1 arbitrarily slowly, proves every almost-integer sequence λ_n = n + α_n (α_n→0, α_n≠0) is p-generating, and exhibits p-generating Λ whose successive differences λ_{n+1}-λ_n attain exactly two distinct positive values. The constructions are presented as sharp relative to the known non-completeness of Hadamard-lacunary sets and arithmetic progressions.
What carries the argument
p-generating set: a sequence Λ of real numbers such that the integer translates of some fixed g span L^p(R)
If this is right
- A p-generating set need not satisfy any fixed lower bound on the growth rate of consecutive ratios.
- Perturbing the integers by a null sequence that never hits zero still yields a p-generating set.
- Sets of finite local complexity (only two distinct gaps) can be p-generating.
- The arithmetic obstructions that block Hadamard-lacunary or purely arithmetic sets do not apply to these milder irregularities.
Where Pith is reading between the lines
- The same families might remain p-generating when the underlying measure is replaced by a non-Lebesgue weight, though the paper does not address weighted spaces.
- One could test whether the two-difference construction extends to sequences whose gaps take finitely many (rather than exactly two) values.
- The almost-integer result suggests that p-generating sets are stable under small perturbations of arithmetic progressions, a stability that may hold in other function spaces.
Load-bearing premise
That for each constructed family there exists at least one function g whose translates by the points of Λ span the whole space L^p(R).
What would settle it
An explicit counterexample showing that, for some 1<p≤2 and one of the three families, no g has translates that span L^p(R), or a direct verification that a specific almost-integer sequence fails to be p-generating.
read the original abstract
A real sequence $\Lambda = \{\lambda_n\}_{n=1}^\infty$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-\lambda_n)\}_{n=1}^\infty$ 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 $\Lambda$ of positive real numbers can be very sparse, namely, the ratios $\lambda_{n+1} / \lambda_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every "almost integer" sequence $\Lambda$, i.e. satisfying $\lambda_n = n + \alpha_n$, $0 \neq \alpha_n \to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $\Lambda$ such that the successive differences $\lambda_{n+1} - \lambda_n$ attain only two different positive values. The constructions are, in a sense, sharp: it is well known that $\Lambda$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.
Editorial analysis