arXiv subjects

Bryce Kerr

Publications and source records attributed to Bryce Kerr.

40 records · Page 3Linked to original sources

arXiv2013

Rational exponential sums over the divisor function

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.

math.NT↗

arXiv2013

Solutions to polynomial congruences in well shaped sets

We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when the solutions lie in a very general class of sets, including all convex sets.

math.NT↗

arXiv2013

Incomplete exponential sums over exponential functions

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2πiag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration.

math.NT↗

arXiv2012

On the Distribution of Values and Zeros of Polynomial Systems over Arbitrary Sets

Let $G_1,..., G_n \in \Fp[X_1,...,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\Fp$ of $p$ elements. A result of {É}. Fouvry and N. M. Katz shows that under some natural condition, for any fixed $\varepsilon$ and sufficiently large prime $p$ the vectors of fractional parts $$ (\{\frac{G_1(\vec{x})}{p}},...,\{\frac{G_n(\vec{x})}{p}}), \qquad \vec{x} \in Γ, $$ are uniformly distributed in the unit cube $[0,1]^n$ for any cube $Γ\in [0, p-1]^m$ with the side length $h \ge p^{1/2} (\log p)^{1 + \varepsilon}$. Here we use this result to show the above vectors remain uniformly distributed, when $\vec{x}$ runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences.

math.NT↗