98 citations · 123 across the 7 of their papers we have counts for
8 papers
Diophantine approximation on curves and the distribution of rational points: divergence theory
V. Beresnevich, R. C. Vaughan, S. Velani +1
In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying a…
Diophantine approximation on manifolds and lower bounds for Hausdorff dimension
Victor Beresnevich, Lawrence Lee, Robert C. Vaughan +1
Given and , let denote the classical set of -approximable points in , which consists of th…
The Least Number with Prescribed Legendre Symbols
Brandon Hanson, Robert C. Vaughan, Ruixiang Zhang
In this article we estimate the number of integers up to which can be represented by a positive-definite, binary integral quadratic form of discriminant which is small relative…
Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory
Victor Beresnevich, Robert C. Vaughan, Sanju Velani +1
In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \…
The asymptotic formula in Waring's problem: higher order expansions
Robert C. Vaughan, Trevor D. Wooley
When k > 1 and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum o…
Mean value theorems for binary Egyptian fractions II
Jing-Jing Huang, Robert C. Vaughan
In this article, we continue with our investigation of the Diophantine equation and in particular its number of solutions for fixed . We pro…
Mean Value Theorems for Binary Egyptian Fractions
Jing-Jing Huang, Robert C. Vaughan
In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation , in the case when is fixed and…
Diophantine approximation on planar curves: the convergence theory
R. C. Vaughan, S. L. Velani
The convergence theory for the set of simultaneously -approximable points lying on a planar curve is established. Our results complement the divergence theory developed in `Diop…