arXiv subjects

Shuntaro Yamagishi

Publications and source records attributed to Shuntaro Yamagishi.

22 records · Page 2Linked to original sources

arXiv2015

A generalization of a theorem of Erdős-Rényi to $m$-fold sums and differences

Let $m\geq 2$ be a positive integer. Given a set $E(ω)\subseteq \mathbb{N}$ we define $r_{N}^{(m)}(ω)$ to be the number of ways to represent $N\in \mathbb{Z}$ as any combination of sums $\textit{ and }$ differences of $m$ distinct elements of $E(ω)$. In this paper, we prove the existence of a "thick" set $E(ω)$ and a positive constant $K$ such that $r_{N}^{(m)}(ω)<K$ for all $N\in \mathbb{Z}$. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.

math.NT↗

arXiv2015

The asymptotic formula for Waring's problem in function fields

Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that for all $s \geq t_0$, one has the expected asymptotic formula in Waring's problem over $\mathbb{F}_q[t]$ concerning sums of $s$ $k$-th powers of polynomials in $\mathbb{F}_q[t]$. For each $k$ not divisible by $p$, we derive a minor arc bound from Vinogradov-type estimates, and obtain bounds on $\widetilde{G}_q(k)$ that are quadratic in $k$, in fact linear in $k$ in some special cases, in contrast to the bounds that are exponential in $k$ available only when $k < p$. We also obtain estimates related to the slim exceptional sets associated to the asymptotic formula.

math.NT↗

arXiv2015

Diophantine approximation of polynomials over $\mathbb{F}_q[t]$ satisfying a divisibility condition

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over $\mathbb{F}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.

math.NT↗

arXiv2013

Computing the moment polynomials of the zeta function

We describe a method to accelerate the numerical computation of the coefficients of the polynomials $P_k(x)$ that appear in the conjectured asymptotics of the $2k$-th moment of the Riemann zeta function. We carried out our method to compute the moment polynomials for $k \leq 13$, and used these to experimentally test conjectures for the moments up to height $10^8$.

math.NT↗