Zobrazeno 1 - 10
of 183
pro vyhledávání: '"Brüdern Jörg"'
Autor:
Brüdern Jörg, Robert Olivier
Publikováno v:
Open Mathematics, Vol 22, Iss 1, Pp 195-202 (2024)
Asymptotic formulae are established for the number of natural numbers mm with largest square-free divisor not exceeding mϑ{m}^{{\vartheta }}, for any fixed positive parameter ϑ{\vartheta }. Related counting functions are also considered.
Externí odkaz:
https://doaj.org/article/c0f4f005ab91415d9b2be8d154347e40
If $\mathscr A$ is a set of natural numbers of exponential density $\delta$, then the exponential density of all numbers of the form $x^3+a$ with $x\in\mathbb N$ and $a\in\mathscr A$ is at least $\min(1, \frac 13+\frac 56 \delta)$. This is a consider
Externí odkaz:
http://arxiv.org/abs/2409.16795
Autor:
Bruedern, Joerg, Wooley, Trevor D.
We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of $\alpha n^k$. In particular, when $k\ge 6$ and $\rho(k)$ is defined via the relation $\rho(k)^{-1}=k(\log k+8.0
Externí odkaz:
http://arxiv.org/abs/2408.06441
Autor:
Bruedern, Joerg, Wooley, Trevor D.
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 positive intege
Externí odkaz:
http://arxiv.org/abs/2402.09537
Autor:
Brüdern, Jörg, Robert, Olivier
Fix a positive real number $\theta$. The natural numbers $m$ with largest square-free divisor not exceeding $m^\theta$ form a set $\mathscr{A}$, say. It is shown that whenever $\theta>1/2$ then all large natural numbers $n$ are the sum of two element
Externí odkaz:
http://arxiv.org/abs/2306.12431
Autor:
Brüdern, Jörg, Robert, Olivier
Asymptotic formulae are established for the number of natural numbers $m$ with largest square-free divisor not exceeding $m^{\vartheta}$, for any fixed positive parameter $\vartheta$. Related counting functions are also considered.
Comment: 12 p
Comment: 12 p
Externí odkaz:
http://arxiv.org/abs/2306.05981
Autor:
Bruedern, Joerg, Wooley, Trevor D.
Let $k_i\in \mathbb N$ $(i\ge 1)$ satisfy $2\le k_1\le k_2\le \ldots $. Freiman's theorem shows that when $j\in \mathbb N$, there exists $s=s(j)\in \mathbb N$ such that all large integers $n$ are represented in the form $n=x_1^{k_j}+x_2^{k_{j+1}}+\ld
Externí odkaz:
http://arxiv.org/abs/2302.12920
Autor:
Bruedern, Joerg, Wooley, Trevor D.
We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.
Comment: 37 pages
Comment: 37 pages
Externí odkaz:
http://arxiv.org/abs/2211.10397
Autor:
Bruedern, Joerg, Wooley, Trevor D.
Let $k$ be a natural number and let $c=2.134693\ldots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\infty)$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the number of
Externí odkaz:
http://arxiv.org/abs/2211.10387
Autor:
Bruedern, Joerg, Wooley, Trevor D.
Let $G(k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$-th powers. Then for all $k\in \mathbb N$, one has \[ G(k)\le \lceil k(\log k+4.20032)\rceil . \] O
Externí odkaz:
http://arxiv.org/abs/2211.10380