Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2112.11813

Weak Poissonian Correlations

Manuel Hauke·Agamemnon Zafeiropoulos
View original source ↗Read source document ↗Export metadata ↓

Abstract

We examine a property of sequences called Poissonian pair correlations with parameter $0\leqslant β\leqslant 1$ (abbreviated as $β$-PPC). We prove that when $β<1,$ the property of $β$-PPC, also known as weak Poissonian correlations, can be detected at the behaviour of sequences at small scales, and show that this does not happen for the classical notion of PPC, that is, when $β= 1$. Furthermore, we show that whenever $0\leqslant α< β\leqslant 1$, $β$-PPC is stronger than $α$-PPC. We also include a discussion on weak Poissonian correlations of higher orders, showing that for $β< 1$, Poissonian $β$-correlations of order $k+1$ imply Poissonian $β$-correlations of $k$-th order with the same parameter $β$.

Explore related subjects

math.NTmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Manuel Hauke, Agamemnon Zafeiropoulos. 2021-12-22. Weak Poissonian Correlations. https://arxiv.org/abs/2112.11813

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Real quadratic base changes for $\mathrm{GL}_3$ and integral periods relations

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel's base change to some real quadratic field $E$. This generalizes earlier works of Tilouine-Urban and of Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods, defined using the middle degree of the cuspidal cohomology of $\mathrm{GL}_3(E)$, instead of the top or bottom degrees. We also investigate the Rogawski's stable base change from the quasi-split unitary group $U_E$ associated with $E$ to $\mathrm{GL}_3(E)$. In this situation, we also obtain some results toward a $p$-adic divisibility of automorphic periods.

math.NT↗