In their previous work devoted to the distribution of rational points near Brownian motion, the authors conjectured the existence of an \emph{oscillation principle} governing the asymptotic behavior of the number of rational points with bounded denomi\-nators near the graph of a monofractal curve. In this note, a weaker form of this conjecture is shown to hold for a broad class of deterministic fractal curves. These include the classical Takagi and Weierstrass nowhere differentiable functions, and indeed a prevalent (i.e.~"large") class of functions among those which are H\"older continuous with a given exponent of regularity. This constitutes the first instance of deterministic fractal curves for which a precise count of the rational points under consideration is established.
We prove a fractal analogue of the weighted theorems of Khintchine and Schmidt for products of self-similar measures on the real line whose defining iterated function systems share a common contraction ratio. The proof translates the counting problem into homogeneous dynamics and builds on recent work of B\'enard, He a...
Consider a holomorphic family (fλ)λ∈Λ$(f_\lambda)_{\lambda \in \Lambda }$ of polynomial maps on C$\mathbb {C}$ with the property that a critical point of fλ$f_\lambda$ is persistently preperiodic to a repelling periodic point of fλ$f_\lambda$ . Let Ω$\Omega$ be a bounded stable component of Λ$\Lambda$ with the property...
F. Bianchi, Y. He· Journal of the London Mathem...· 2 citations
We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasov\'a and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depend...
In 1975, Voronin established the celebrated universality theorem for the Riemann zeta-function, and later, Bagchi and Gonek independently obtained its analogue for a certain family of Dirichlet L-functions in the q-aspect. In this paper, we generalize their results by establishing a simultaneous universality theorem fo...
H. Nagoshi· Journal of Mathematics and S...· 0 citations
We prove the existence of an essential spectral gap for convex cocompact complex hyperbolic manifolds under the hypothesis that the limit set does not contain a complex circle. We apply an extension to complex hyperbolic manifolds due to Quan of the approach of Vasy to meromorphic continuation of the resolvent. This al...
The dynamic of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is determined by the forward orbits of its critical points. Such a map is called {\em postcritically finite} if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle. Thes...
Edgar Saenz, Dheemanth Samji· 0 citations
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