arXiv subjects
Publications and source records attributed to Robert C. Vaughan.
This memoir is a survey of theorems and inequalities which have grown out of, and extended, the seminal estimate of Montgomery \cite{HM70} \begin{multline*} V(x,Q)=\sum_{q\le Q}\sum_{\substack{a=1\\ (a,q)=1}}^q \left| ψ(x;q,a) - \frac{x}{ϕ(q)} \right|^2 \\ = Qx\log x + \textstyle O\big(Qx\log\frac{2x}{Q}\big) + O\big(x^2(\log x)^{-A}\big)., \end{multline*}
We obtain asymptotics for sums of the form $$ \sum_{n=1}^P e(α_kn^k + α_1n), $$ involving lower order main terms. As an application, we show that for almost all $α_2 \in [0,1)$ one has $$ \sup_{α_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(α_1(n^3+n) + α_2 n^3) \Big| \ll P^{3/4 + \varepsilon}, $$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.
Given $n\in\mathbb{N}$ and $τ>\frac1n$, let $\mathcal{S}_n(τ)$ denote the classical set of $τ$-approximable points in $\mathbb{R}^n$, which consists of ${\bf x}\in \mathbb{R}^n$ that lie within distance $q^{-τ-1}$ from the lattice $\frac1q\mathbb{Z}^n$ for infinitely many $q\in\mathbb{N}$. In pioneering work, Kleinbock $\&$ Margulis showed that for any non-degenerate submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $τ>\frac1n$ almost all points on $\mathcal{M}$ are not $τ$-approximable. Numerous subsequent papers have been geared towards strengthening this result through investigating the Hausdorff measure and dimension of the associated null set $\mathcal{M}\cap\mathcal{S}_n(τ)$. In this paper we suggest a new approach based on the Mass Transference Principle, which enables us to find a sharp lower bound for $\dim \mathcal{M}\cap\mathcal{S}_n(τ)$ for any $C^2$ submanifold $\mathcal{M}$ of $\mathbb{R}^n$ and any $τ$ satisfying $\frac1n\leτ<\frac1m$. Here $m$ is the codimension of $\mathcal{M}$. We also show that the condition on $τ$ is best possible and extend the result to general approximating functions.
In this article we estimate the number of integers up to $X$ which can be represented by a positive-definite, binary integral quadratic form of discriminant which is small relative to $X$. This follows from understanding the vector of signs when computing the Legendre symbol of small integers $n$ at multiple primes.
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 $M \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $M$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.
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 of s positive integral k-th powers.
For fixed integer $a\ge3$, we study the binary Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular the number $E_a(N)$ of $n\le N$ for which the equation has no positive integer solutions in $x, y$. The asymptotic formula $$E_a(N)\sim C(a) \frac{N(\log\log N)^{2^{m-1}-1}}{(\log N)^{1-1/2^m}}$$ as $N$ goes to infinity, is established in this article, and this improves the best result in the literature dramatically. The proof depends on a very delicate analysis of the underlying group structure.
In this article, we continue with our investigation of the Diophantine equation $\frac{a}n=\frac1x+\frac1y$ and in particular its number of solutions $R(n;a)$ for fixed $a$. We prove a couple of mean value theorems for the second moment $(R(n;a))^2$ and from which we deduce $\log R(n;a)$ satisfies a certain Gaussian distribution with mean $\log 3\log\log n$ and variance $(log 3)^2\log\log n$, which is an analog of the classical theorem of Erd\H os and Kac. And finally these results in all suggest that the behavior of $R(n;a)$ resembles the divisor function $d(n^2)$ in various aspects.
In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.
The inhomogeneous metric theory for the set of simultaneously $ψ$-approximable points lying on a planar curve is developed. Our results naturally incorporate the homogeneous Khintchine-Jarnik type theorems recently established in [Ann. of Math. (2), 166 (2007), pp. 367-426] and [Invent. Math., 166 (2006), pp. 103-124]. The key lies in obtaining essentially the best possible results regarding the distribution of `shifted' rational points near planar curves.