This is a survey of the so-called “quantitative nondivergence” approach to metric Diophantine approximation developed approximately 10 years ago in my collaboration with Margulis. The goal of this paper is to place the theory of approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on Rn. The correspondence betwee...

For diophantine approximation of algebraic real numbers by rationals, Roth’s celebrated theorem settles the issue of the approximation exponent completely in the number field case. But in spite of strong analogies between number fields and function fields, its naive analogue fails in function fields of finite characteristic, and the situation is not even conjecturally understood. In this short ...

– In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associate...

In Diophantine Approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the inhomogeneous exponent of approximation to a generic point in R by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent...

This paper presents and analyzes cryptanalytic attacks on knapsack public key cryptosystems that are based on ideas from Diophantine approximation. Shamir’s attack on the basic Merkle-Hellman knapsack cryptosystem is shown to depend on the existence of ‘‘unusually good’’ simultaneous Diophantine approximations to a vector constructed from the public key. This aspect of Shamir’s attack carries o...

Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s ...

The metrical theory of Diophantine approximation for quaternions is developed using recent results in the general theory. In particular, Quaternionic analogues of the classical theorems of Khintchine, Jarnı́k and Jarnı́k-Besicovitch are established. Introduction Diophantine approximation begins with a more quantitative understanding of the density of the rationals Q in the reals R. The starting p...