Dirichlet's theorem is a fundamental result in metric Diophantine approximation. The refinement of its asymptotic setting yields the psi-well-approximable set (via continued fractions) Kappa(psi) := {x is an element of [0, 1): a(n+1) (x) >= (q(n)(x)) for infinitely many n is an element of N}, and the refinement of its uniform setting leads to the psi-Dirichlet non-improvable set G(psi) := {x in [0, 1) : a(n)(x)a(n+1) (x) >= psi(q(n)(x)) for infinitely many n is an element of N}. where a(n)(x) denotes the n-th partial quotient of x and q(n)(x) is the denominator of the n-th convergent. Surprisingly, as a proper subset of G(psi), the set K(psi) shares the same s-dimensional Hausdorff measure as G(psi). Furthermore, the difference set G(psi) \ K(psi) has the same Hausdorff dimension as K(psi). Therefore, to gain deeper insights into these results, this paper investigates how the growth rate of the product of partial quotients a(n)(x)a(n+1) (x) affects the Hausdorff dimension relative to that of individual partial quotients a(n)(x). More precisely, for any alpha, beta, gamma is an element of [0, infinity), we investigate the Hausdorff dimension of the intersection of D (c) (gamma) = {x is an element of [0, 1) : lim sup(n ->infinity) log a(n)(x)a(n+1)(x)/log q(n)(x) >= gamma} and Epsilon(alpha, beta) = {x is an element of [0, 1) : lim sup(n ->infinity) log a(n+1)(x)/log q(n)(x) = alpha and lim sup(n ->infinity) log a(n+1)(x)/log q(n)(x) = beta}.
In this article, we introduce the finite element method to analyze a membrane with a Koch snowflake-shaped boundary. The fractal nature of this domain presents unique challenges due to its intricate boundary structure. Our approach involves discretizing the domain, estimating the error, and proving convergence. With these aspects addressed, we solve a shape optimization problem to determine the optimal thickness of the membrane. These findings provide valuable insights into how fractal boundaries affect structural performance and optimization.
We establish explicit decay rates for the Fourier transforms of self-similar probability measures supported on non-singleton self-similar sets, under a non-Liouville condition on a logarithmic ratio of contraction ratios. The decay exponent is given in closed form as a function of any chosen upper regularity exponent and the non-Liouville degree, turning the implicit bounds of Li-Sahlsten into fully effective estimates. In the presence of pairwise disjoint images, we obtain explicit decay for arbitrary self-similar measures, and identify a canonical one achieving the maximal decay permitted by our method. We then apply the theory to L & uuml;roth digit-restricted sets, deriving computable exponents from explicit lower bounds for linear forms in logarithms and giving a worked numerical example for the two-digit case.
For a sequence of complex parameters \omega=(c_{m}) we consider the non-autonomous composition of functions P_{c_m}(z) = z^{n}+c_{m} z^{k} with n>k\geq 2 . The definitions of Julia, \mathcal{J}_{\omega} , and Fatou, \mathcal{F}_{\omega} , sets are naturally generalized to this setting. In this paper, we study the connectedness and some dynamic properties of the Julia set provided that the sequence \omega is randomly chosen from some bounded Borel set W\subset \mathbb{C} .
We show that all Cantor sets in \mathbb{R}^{d} can be accompanied by another Cantor set in \mathbb{R}^{d} so that their product has a pinned tree distance set with nonempty interior. As a corollary, we construct Cantor sets of Hausdorff dimension d/2 in \mathbb{R}^{d} for even d that have a pinned tree distance set with nonempty interior. Our results generalize to the setting in which the Euclidean distance, |x-y| , is replaced by a general function, \phi(x,y) , satisfying a mild derivative condition.
