Related papers: Construction of a one-dimensional set which asympt…

We analytically compute asymptotic expansions of a 1-dimensional sub-manifold of stable and unstable manifolds in a 4-dimensional symplectic mapping by using the method called asymptotic expansions beyond all orders. This method enables us…

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale…

We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are…

Let d be a squarefree integer. Does there exist four squares in arithmetic progression over Q(sqrt{d})? We shall give a partial answer to this question, depending on the value of d. In the affirmative case, we construct explicit arithmetic…

We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the…

This paper investigates the analytic and structural properties of the $\phi$-lower Assouad dimension, a generalized notion extending the lower Assouad dimension. We establish the equivalence of $\phi$-lower Assouad dimensions with respect…

Let F be a fixed finite field of characteristic at least 5. Let G = F^n be the n-dimensional vector space over F, and write N := |G|. We show that if A is a subset of G with size at least c_F N(log N)^{-c}, for some absolute constant c > 0…

We introduce a probability distribution on $\mathcal{P}([0,1]^d)$, the space of all Borel probability measures on $[0,1]^d$. Under this distribution, almost all measures are shown to have infinite upper quasi-Assouad dimension and zero…

We study the distribution of partition parts in arithmetic progressions and find asymptotic results that capture all exponentially growing terms. This is accomplished by studying the behavior of non-modular Eisenstein series that appear in…

We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of `passing to weak tangents'. First, we solve an analogue of Falconer's distance set…

Furstenberg, Glasscock, Bergelson, Beiglboeck have been studied abundance in arithmatic progression on various large sets like piecewise syndetic, central, thick, etc. but also there are so many sets in which abundance in progression is…

The connections between quasi-Assouad dimension and tangents are studied. We apply these results to the calculation of the quasi-Assouad dimension for a class of planar self-affine sets. We also show that sets with decreasing gaps have…

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by the partial sums of a…

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar…

Given a non-negative, decreasing sequence $a$ with sum $1$, we consider all the closed subsets of $[0,1]$ such that the lengths of their complementary open intervals are given by the terms of $a$, the so-called complementary sets. In this…

We show that if a finite, large enough subset A of an arbitrary abelian group satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|, then A must contain a three-term arithmetic progression whose terms are not all…

In this paper we study the Assouad dimension of graphs of certain L\'evy processes and functions defined by stochastic integrals. We do this by introducing a convenient condition which guarantees a graph to have full Assouad dimension and…

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a…

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of "structured" $\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in…