Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.23401

Extreme values and logarithm laws for toral translations

Sourav Das·Anish Ghosh·Sundara Narasimhan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study extreme values of normalized $r$-th nearest neighbour distance functions in inhomogeneous Diophantine approximation, proving Weibull and Fréchet limit laws when the shift satisfies a natural Diophantine condition. Using these extreme value laws we prove corresponding logarithm laws for these functions. Our approach is based on homogeneous dynamics. The key input is to adapt the Poisson approximation strategy of Björklund and Gorodnik (arXiv:2201.05116) combined with an effective multiple equidistribution result, which we deduce using results of Strömbergsson (arXiv:1309.6103) and of Björklund and Gorodnik (arXiv:2105.05468).

Explore related subjects

math.NTmath.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sourav Das, Anish Ghosh, Sundara Narasimhan. 2026-08-24. Extreme values and logarithm laws for toral translations. https://arxiv.org/abs/2608.23401

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↗