Related papers: Bounds for the Cubic Weyl Sum

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by…

Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…

For positive integers $k$ and $n$ let $\sigma_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erd\H{o}s and Kac asked whether, for every $k$, the number $\alpha_k = \sum_{n\geq 1} \frac{\sigma_k(n)}{n!}$ is irrational. It…

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $\alpha x^3+\beta x^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine…

A strong error estimate for the uniform rational approximation of $x^\alpha$ on $[0,1]$ is given, and its proof is sketched. Let $E_{nn}(x^\alpha,[0,1])$ denote the minimal approximation error in the uniform norm. Then it is shown that…

We study exponential sums of the form $\sum_{n=1}^N e^{2\pi i a b^n/m}$ for non-zero integers $a,b,m$. Classically, non-trivial bounds were known for $N\ge \sqrt{m}$ by Korobov, and this range has been extended significantly by Bourgain as…

In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for…

We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums which for a given $\alpha \in (1/2,1)$ achieve the order at least $N^{\alpha}$ for infinitely many sum lengths $N$. For Weyl sums with polynomials…

Let $b > a > 0$. We prove the following asymptotic formula $$\sum_{n\ge 0} \big\lvert\{x/(n+a)\} - \{x/(n+b)\}\big\rvert = \frac{2}{\pi}\zeta(3/2)\sqrt{cx} + O(c^{2/9}x^{4/9}),$$ with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…

Let $\pi_1, \pi_2, \pi_3$ be three cuspidal automorphic representations for the group ${\rm SL}(2, \Bbb{Z})$, where $\pi_1$ and $\pi_2$ are fixed and $\pi_3$ has large conductor. We prove a subconvex bound for $L(1/2, \pi_1 \otimes \pi_2…

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq…

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n\leq x}f(n)e(n\alpha)$ where $f$ is a $1$-bounded multiplicative function and $\alpha\in\mathbb R$, close to the conjectured $\ll \frac{x}{\sqrt{q}}+…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

We show that the equation $\lambda_1 n_1^2 + ... + \lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\log N)^{-c_s}$, provided that $s \geq 7$ and the coefficients $\lambda_i$ sum to zero and satisfy…

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$…

In a previous paper with Schmid (math.NT/0402382) we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting,…

In this article, we prove that the double inequality $$\alpha G(a,b)+(1-\alpha)C(a,b)<M(a,b)<\beta G(a,b)+(1-\beta)C(a,b)$$ holds true for all $a,b>0$ with $a\neq b$ if and only if $\alpha\geq 5/9$ and $\beta\leq…

The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-\epsilon}$ for any $\epsilon > 0$. Using the optimized methods in a recent…