Gap statistics and higher correlations for geometric progressions modulo one

On the number of gaps of sequences with Poissonian pair correlations

“…For each 1) be the piecewise linear transformation defined as follows: if ρ 1 (ℓ) = min{µ 1 (ℓ), ν 1 (ℓ)}, then f ℓ is the transformation that maps…”

“…Note how, heuristically, f (1) considers the intervals that arise from the points with index at most N 1 , and then exchanges an interval in the left half of [0, 1] which contains no point with index in (N 1 , 2N 1 ] against an equal-length interval in the right half of [0, 1] that does contain one or more points with index in (N 1 , 2N 1 ]; thus, f (1) effectively shifts points from the right half of [0, 1] into the left half without changing the overall gap statistic. Note also how this procedure requires the existence of gaps of the same size, which can be exchanged against each other; in order to have a significant overall effect, leading to a non-uniform distribution of the resulting sequence (x n ) n∈N , we need to ensure that there are "many" gaps of equal size to which the procedure can be applied.…”

“…Rudnick, Sarnak and Zaharescu [22] gave a Diophantine criterion for α which guarantees that the sequence ({n 2 α}) n∈N has exponential gaps (without providing an explicit example of a number α satisfying the condition), and Lutsko and Technau proved that the sequence ({α(log n) A }) n∈N for α > 0 and A > 1 has exponential gaps [16]. Among metric results, it is known that the sequence ({q n α}) n∈N for a lacunary (q n ) n∈N has exponential gap distribution for almost all α [7,23], and that ({α n }) n∈N has exponential gaps for almost all α > 1 [1].…”

“…Applying Theorem 1 to a sequence (x n ) n∈N that is uniformly distributed and has distinct gaps, we see that any sequence (y n ) n∈N with the same gaps is automatically uniformly distributed as well. It might be tempting to think that the assumption of "same gap structure and distinct gaps" in Theorem 1 can be replaced by "same gap distribution" alone, so that having the same gap distribution already implies (1). Since a random sequence has exponential gaps and is equidistributed almost surely, in view of (1) one might be led to expect that any other sequence having exponential gaps must be equidistributed as well.…”

The distribution of spacings of real‐valued lacunary sequences modulo one

“…Rudnick and Zaharescu proved [15] that if $$(a_{n})$$ is a lacunary sequence of integers , then the nearest‐neighbour spacing distribution of $$(\alpha a_{n})$$ is Poissonian for almost all $$\alpha \in \mathbb {R};$$ as will be detailed below, the main goal of this note is to show an analogous result for real‐valued lacunary sequences. Another natural question about real‐valued lacunary sequences with a different notion of randomization (which dramatically changes the problem) is whether the sequence $$(\alpha ^{n})$$ has a Poissonian nearest‐neighbour spacing distribution for almost all $$\alpha >1$$ — this was recently answered positively in [1], as a special case of a more general family of sequences (which include some sub‐lacunary sequences as well) having this property. Very little is known for polynomially growing sequences.…”