Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.13099

Khintchine dichotomy for images of self-similar measures and almost random walks

Liyang Shao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Following the celebrated breakthrough in Bénard-He-Zhang, we prove Khintchine dichotomy for push-forwards of self-similar measures on $\mathbb{R}^d$ under uniform $C^{1+α}$ maps. For this, we also study almost random walks on $\mathrm{SL}(d+1,\mathbb{R})/\mathrm{SL}(d+1,\mathbb{Z})$ and prove their effective equidistribution.

Explore related subjects

math.DSmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Liyang Shao. 2026-10-09. Khintchine dichotomy for images of self-similar measures and almost random walks. https://arxiv.org/abs/2610.13099

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

KEEP EXPLORING

Related papers

arXiv2026

Survivor Set of Gauss Map with a Hole at $1$

Let $T$ be the Gauss map. For each $α\in [0,1]$, let $E(α)$ be the survivor set consisting of all $x\in [0,1]$ whose orbit $\{T^k(x):k\ge 0\}$ never enters the interval $(α,1]$. We prove that the set-valued function $α\mapsto E(α)$ is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function $α\mapsto E(α)$ remains constant. We also describe the points at which the map $α\mapsto E(α)$ is not locally constant. Further, we obtain the Hausdorff dimension of $E(α)$ for all $α\in [0,1]$. We prove that the function $α\mapsto \dim_H(E(α))$ forms a devil's staircase on $(0,1]$, but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical $α$-continued fraction expansion and regular continued fraction expansion.

math.DS↗
arXiv2026

Decomposition results for multiplicative actions and applications

Motivated by partition regularity problems of homogeneous quadratic equations, we prove multiple recurrence and convergence results for multiplicative measure preserving actions with iterates given by rational sequences involving polynomials that factor into products of linear forms in two variables. We focus mainly on actions that are finitely generated, and the key tool in our analysis is a decomposition result for any bounded measurable function into a sum of two components, one that mimics concentration properties of pretentious multiplicative functions and another that mimics vanishing properties of aperiodic multiplicative functions. Crucial to part of our arguments are some new seminorms that are defined by a mixture of addition and multiplication of the iterates of the action, and we prove an inverse theorem that explicitly characterizes the factor that these seminorms naturally determine.

math.DS↗
arXiv2026

Shadowing in CR-Dynamical Systems

A CR-dynamical system is a pair $(X, G)$, where $X$ is a compact metric space and $G$ is a closed relation on $X$. We introduce four $(i,j)$-shadowing properties, obtained by independently varying the quantifiers in the definitions of pseudo-orbit and shadowing. These properties coincide with the classical shadowing property for single-valued systems, while $(2, 2)$-shadowing agrees with the established notion of shadowing for set-valued dynamical systems. We investigate how the classical characterisations of shadowing for the identity map, periodic maps and isometries extend to closed relations. In particular, we obtain rigidity results for $(2, 1)$-shadowing and positive results for $(1, 2)$-shadowing on totally disconnected spaces. Finally, whenever the infinite Mahavier product is non-empty, we prove that a closed relation has the $(2, 2)$-shadowing property if, and only if, its associated Mahavier dynamical system has the shadowing property. This removes the surjectivity hypothesis from a recent corresponding result for set-valued dynamical systems.

math.DS↗