Related papers: Generalized Diophantine Approximation on Higher-Di…
We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…
Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $\Gamma < G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the…
The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…
For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by a finite descent obstruction, in the sense that the rational…
We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue…
We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…
Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, with smooth generic fiber $X_K$. We give a formula that relates the dimension of the first Arakelov-Chow vector space of $X$ with the Mordell-Weil rank of the…
Consider the classical problem of rational simultaneous approximation to a point in $\mathbb{R}^{n}$. The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and…
The inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the theory is established for all such manifolds. Our…
It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum…
We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this…
We use the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite…
We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group,…
The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…
Without assuming the Northcott property we provide an upper bound on the number of "big solutions" of a special system of Diophantine inequalities over proper adelic curves. This system is interesting since it is a stronger version Roth's…
We prove a uniform version of the Dynamical Mordell-Lang Conjecture for \'etale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined…
We present a hypergeometric construction of rational approximations to $\zeta(2)$ and $\zeta(3)$ which allows one to demonstrate simultaneously the irrationality of each of the zeta values, as well as to estimate from below certain linear…
The aim of the paper is twofold. Firstly, by using the constant rank level set theorem from differential geometry, we establish sharp upper bounds for the dimensions of the solution sets of polynomial variational inequalities under mild…
We associate to a filtration of a graded linear series of a big line bundle a concave function on the Okounkov body whose law with respect to Lebesgue's measure describes the asymptotic distribution of the jumps of the filtration. As a…
We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of…