Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.37119

Fractal transference principle for conformal iterated function systems

Yuto Nakajima·Hiroki Takahasi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish a complex one-dimensional fractal transference principle connecting density combinatorics with the fractal geometry of complex continued fractions. Our framework applies to a broad class of conformal iterated function systems indexed by lattice points, including those associated with the Hurwitz continued fraction, and transfers positive-density combinatorial patterns to digit sets while preserving the maximal Hausdorff dimension. As applications, we obtain fractal counterparts of the multidimensional and polynomial Szemerédi theorems, as well as their relativized versions yielding a counterpart of the multidimensional Green--Tao theorem.

Explore related subjects

math.DSmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuto Nakajima, Hiroki Takahasi. 2026-10-27. Fractal transference principle for conformal iterated function systems. https://arxiv.org/abs/2610.37119

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

KEEP EXPLORING

Related papers

arXiv2026

Infinite unrestricted sumsets in subsets of abelian groups with large density

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect to an appropriate Følner sequence, so that $A$ contains a sumset of the form $\{t+b_1+b_2\colon b_1,b_2\in B\}$ or $\{b_1+b_2\colon b_1,b_2\in B\}$, for some infinite $B\subset G$ and some $t\in G$. Both assumptions on $G$ are necessary for our results to be true. We also characterize the Følner sequences for which this is possible. Finally, we show that our lower bounds are optimal in a strong sense.

math.DS↗
arXiv2026

Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization.

math.DS↗
arXiv2026

Solvable Lie Foliations on Compact Five-Manifolds: Classification, Cohomological Obstruction, and Non-Homogeneous Examples

We study $\GA$-Lie foliations on compact five-manifolds. We establish a structural classification of the holonomy group $Γ$ into five types according to its algebraic structure inside $\GA$. For three of these types (I, II, IIIa) we prove homogeneity via Meigniez's theorem and the Matsumoto--Tsuchiya theorem. For Type~IIIb we prove non-homogeneity by exhibiting a non-polycyclic holonomy group, extending the Dathe--Ndiaye construction \cite{DN11}. We introduce a cohomological class $[F]\in H^2(Γ^{\mathrm{ab}},[Γ,Γ])$ as a \emph{necessary} obstruction to homogeneity: if $\mathcal{F}$ is homogeneous and $Γ$ is polycyclic then $[F]=0$. We show that Type IIIb foliations satisfy $[F]\neq 0$. For Type IIIc, we prove that certain foliations are non-homogeneous by a direct argument when the generating translations are $\Q(λ)$-linearly independent; we exhibit explicit homogeneous examples showing that this type is \emph{not} universally non-homogeneous. Finally, we construct the first known non-homogeneous $\mathrm{Sol}$-Lie foliation in dimension five by a verified lattice deformation.

math.DS↗