Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1510.08481

Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits

Ilya Khayutin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates and homogeneous dynamics.

Explore related subjects

math.NTmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ilya Khayutin. 2019-02-25. Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits. https://doi.org/10.1215/00127094-2019-0016

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

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.

math.NT↗