Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.31515

On the asymptotic average diameter of blocks of uniformly distributed sequences and related results

Sebastian Heintze·Wolfgang Trutschnig
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper was triggered by recent results on the maximal `average distance between consecutive points' of uniformly distributed sequences (u.f.d.s.). Here we address a generalized version of this question, consider pairwise maximal/minimal/total distances in blocks/segments of $ d \geq 2 $ consecutive points of u.f.d.s., and derive sharp upper bounds for all three aggregations. Our main idea of proof consists in, firstly, adding degrees of freedom, secondly, translating the resulting problem to a solvable optimization problem over the compact family of $ d $-stochastic measures, and, thirdly, showing that the obtained bounds are also sharp bounds for the original problem.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sebastian Heintze, Wolfgang Trutschnig. 2026-05-29. On the asymptotic average diameter of blocks of uniformly distributed sequences and related results. https://arxiv.org/abs/2605.31515

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

KEEP EXPLORING

Related papers

arXiv2026

On vector valued automorphic forms for the Weil representation

We develop a theory of vector valued automorphic forms associated to the Weil representation $ω_f$ and corresponding to vector valued modular forms transforming with the ``finite'' Weil representation $ρ_L$. For each prime $p$ we determine the structure of a vector valued spherical Hecke algebra depending on $ω_f$, which acts on the space of automorphic forms.

math.NT↗
arXiv2026

Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Bhargava et al., we prove unconditionally that a positive proportion of imaginary quadratic fields meet our criterion when $p=3$. For $p=11,13,31,37$, the analogous conclusions obtained from the rank-zero twist families of Kriz--Li are conditional on the vanishing of the $p$-primary Tate--Shafarevich groups for a positive relative proportion of those twists.

math.NT↗
arXiv2026

The standard $L$-function attached to a vector valued modular form

We define two $L$-functions associated to a common vector valued eigenform $f$ transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of $f$. The second one can be interpreted as standard $L$-function defined as an Euler product where each $p$-factor is a rational function in terms of two unramified characters of the $p$-adic field $\Q_p$. We show that both $L$-functions are related and prove further that they both can be continued meromorphically to the whole complex $s$-plane.

math.NT↗