Pointwise improvement of the error term in the higher rank hyperbolic lattice counting problem

Dimitrios Chatzakos; Panagiotis Dimakis; Christos Katsivelos

Home/Archive/arXiv

arXiv · 2610.16668

Pointwise improvement of the error term in the higher rank hyperbolic lattice counting problem

Dimitrios Chatzakos·Panagiotis Dimakis·Christos Katsivelos

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove an upper bound for the error term in the higher rank hyperbolic lattice counting problem for ${\hbox{SL}(n, \mathbb{Z})}$. Blomer and Lutsko achieved recently a significant improvement of previous known results by Duke--Rudnick--Sarnak and Gorodnik--Nevo--Yehoshua. We deduce an improved upper bound for the error which holds for almost all pairs $z,w \in {\hbox{SL}(n, \mathbb{R})}$. We also derive a $Ω$-result for the error term, in the spirit of Rudnick--Phillips.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Dimitrios Chatzakos, Panagiotis Dimakis, Christos Katsivelos. 2026-10-12. Pointwise improvement of the error term in the higher rank hyperbolic lattice counting problem. https://arxiv.org/abs/2610.16668

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

KEEP EXPLORING

Related papers

arXiv2026

On sparsity of integral points in orbits and correspondences with big iterated pullbacks

We prove new unconditional results of sparsity of integral points on orbits under many maps and correspondences in arbitrary dimensions, generalizing theorems of Yasufuku(2015) and others. The main ingredients are new diophantine approximation tools and recent constructions for correspondences due to Ingram (2011).

math.NT↗

arXiv2026

Sporadic points on $X_0(N)$

We determine all integers $N$ for which the modular curve $X_0(N)$ admits a sporadic CM point (of any degree), as well as all $N$ for which $X_0(N)$ admits a sporadic point, whether CM or non-CM. In a sense, our results generalize the classification of isogenies of elliptic curves over $\Q$ due to Mazur and Kenku: their work determines the $X_0(N)$ with degree 1 sporadic points, whereas we classify all $X_0(N)$ that have a sporadic point of arbitrary degree.

math.NT↗

arXiv2026

On the Number of Prime Factors of Consecutive Integers

We prove that there are infinitely many $n$ such that $ω(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao and Teräväinen (2025), who have $O(k)$ in place of $O(\log k)$. As corollaries, we make progress on a number of questions posed by Erdős. The proof is based on a quantitative refinement of the Tao-Teräväinen probabilistic argument, combining a refined sieve construction with upper bounds for high moments and the resulting tail estimates. Moreover, we formulate a conjecture on integers with many prime factors based on Cramér-type random models. Assuming this conjecture, the main bound is essentially sharp.

math.NT↗