Let $F(n)$ be the largest size of a set $A\subseteq[1,n]$ such that $a\nmid bc$ whenever $a,b,c\in A$ and $a\notin\{b,c\}$, with $b$ and $c$ allowed to coincide. We prove \[ F(n)=\pi(n)+\left(\frac{27}{2}+o(1)\right)\frac{n^{2/3}}{(\log n)^2}. \] This determines the second-order constant conjectured by Erd\H{o}s; the upper bound keeps the leading constants in his multiplicative basis, while the lower bound packs scale-separated prime triples by proper edge-colourings.
Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=Cx^{1/2}(\log x)^2$ with $C>0$. The sum of $p(n)/n$ over composite $n$ lying in the short interval $[x,\,x+L_C(x)]$, a question raised by Erd\H{o}s and Graham, is studied. (i) The constant $c=8$ in the mean asymptotic is estimated \[ S(x)=\sum_{{n<x,\ n\ \text...
Let $p$ be a prime, $d\ge 2$, $H\in [1,p)$, and $\ln\ln p = o(\ln H)$. We prove that $$ \min_{1\le n \le H} n \left\|\frac{a_1 n}{p}\right\|\ldots \left\|\frac{a_d n}{p}\right\| \approx \frac{1}{(\ln p)^{d-1} \ln H} $$ for"almost all"$a \in ({\Bbb Z} / p{\Bbb Z})^d$.
Let $L$ be a fixed set of positive integers. A family $\mathcal{F}\subseteq 2^{[n]}$ is called $L$-differencing if $\lvert A\setminus B\rvert\in L$ for every ordered pair of distinct members $A,B\in\mathcal{F}$. A longstanding conjecture of Frankl, proposed in 1985, asserts that every $L$-differencing family has size a...
For $n\ge1$, let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain $n$ pairwise distinct integers $a_1, a_2, \dots, a_n$ with $k \mid a_k$ for $1\le k\le n$, and define $h_{\mathbb P}(n)$ analogously for the primes at most $n$. We prove \[ F(n)\le n^{4/3}\exp\!\left(O\!\left(\frac{\log n}{\log\log...
Let $P=\{0,a,b\}$, where $0<a<b$ and $\gcd(a,b)=1$. For a finite set $A\subset\mathbb Z$, let $M_P^+(A)$ count the copies $x,x+ad,x+bd\in A$ with $d>0$, and let $M_P(A)$ count the copies with any $d\ne0$. We prove that every such three-point pattern other than the arithmetic progression $\{0,1,2\}$ satisfies \[ M_P^+(A...
For an odd prime $p$, let $m(p)$ be the minimum cardinality of a set $A\subseteq \mathbb Z/p\mathbb Z$, with $|A|\geq2$, such that no sum in $A+A$ has a unique representation as an unordered pair from $A$, with repetition allowed. Bedert proved \[ m(p)\gg \log p\, \frac{\sqrt{\log^{(3)}p}}{\log^{(4)}p}. \] We prove the...
Jiao-Long Cao, Ye Yuan· 0 citations
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