Zobrazeno 1 - 10
of 96
pro vyhledávání: '"Koukoulopoulos, Dimitris"'
Autor:
Haddad, Tony, Koukoulopoulos, Dimitris
Let $x \ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb Z \cap [1, x]$, and let $(V_1, V_2, \ldots)$ be a Poisson--Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random objects s
Externí odkaz:
http://arxiv.org/abs/2406.09360
We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $\psi:\mathbb{N}\to[0,1/2]$ be a function such that the series $\sum_{q=1}^\infty \varphi(q)\psi(q)/q$ diverges. In addition, given $\a
Externí odkaz:
http://arxiv.org/abs/2404.14628
We give an improved lower bound for the average of the Erd\H{o}s-Hooley function $\Delta(n)$, namely $\sum_{n\le x} \Delta(n) \gg_\varepsilon x(\log\log x)^{1+\eta-\varepsilon}$ for all $x\geqslant100$ and any fixed $\varepsilon$, where $\eta = 0.353
Externí odkaz:
http://arxiv.org/abs/2308.11987
Autor:
Koukoulopoulos, Dimitris, Tao, Terence
The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u
Externí odkaz:
http://arxiv.org/abs/2306.08615
Covering systems were introduced by Erd\H{o}s in 1950. In the same article where he introduced them, he asked if the minimum modulus of a covering system with distinct moduli is bounded. In 2015, Hough answered affirmatively this long standing questi
Externí odkaz:
http://arxiv.org/abs/2212.01299
Autor:
Koukoulopoulos, Dimitris
Given quantities $\Delta_1,\Delta_2,\dots\geqslant 0$, a fundamental problem in Diophantine approximation is to understand which irrational numbers $x$ have infinitely many reduced rational approximations $a/q$ such that $|x-a/q|<\Delta_q$. Depending
Externí odkaz:
http://arxiv.org/abs/2109.11003
Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random according to $\mu
Externí odkaz:
http://arxiv.org/abs/2007.14567
Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any
Externí odkaz:
http://arxiv.org/abs/2006.13356
Publikováno v:
Discrete Anal., 2020:6, 19 pp
The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on averag
Externí odkaz:
http://arxiv.org/abs/1909.13105
Publikováno v:
Inventiones Math. 232 (2023), 1027-1160
We study the extent to which divisors of a typical integer $n$ are concentrated. In particular, defining the Erd\H{o}s-Hooley $\Delta$-function by $\Delta(n) := \max_t \# \{d | n, \log d \in [t,t+1]\}$, we show that $\Delta(n) \geq (\log \log n)^{0.3
Externí odkaz:
http://arxiv.org/abs/1908.00378