Let $c>1$ be irrational. We prove that, for almost every $x>0$ and every $\varepsilon>0$, there are infinitely many triples $(p,r,s)$, with $p$ and $r$ prime and $s$ an integer, for which \[ 0<px-r<p^{-1/3+\varepsilon},\qquad 0<pcx-s<p^{-1/3+\varepsilon}. \] This improves the exponent $1/5$ in the homogeneous-line case of a theorem of Baier and Ghosh. The arithmetic input is the quantitative prime-counting lower bound in Matom\"aki's proof concerning primes in shrinking neighborhoods of an irrational rotation. We obtain a local first-moment lower bound by fixing the prime numerator and applying the prime number theorem to the admissible prime denominators. Two Selberg sieve squares, together with explicit trigonometric interval approximations, give a pointwise upper bound whose error tends to zero in $L^1$ after normalization. A truncation and density-point argument then yields the almost-everywhere conclusion.
Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Erd\H{o}s Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\de...
Let $\Omega=2\int_0^1(1-t^4)^{-1/2}dt$. We prove that, for every real $\nu>2$, there is $Q\geq2$ such that $q^{-\nu}<|\Omega-p/q|$ for every integer $p$ and every integer $q\geq Q$. Consequently $\Omega$ is irrational and its irrationality exponent is two. The proof combines elliptic separated-weight interpolation, ari...
A prime gap $(p_n, p_{n+1})$ is \textit{frangible} if the smallest prime divisor of every integer $m$ with $p_n<m<p_{n+1}$ is strictly less than the size of the gap, $p_{n+1}-p_n$. Gafni and Tao recently proved that the number of frangible gaps up to $X$ is $\ll X/\log^2 X$, answering a question of Erd\H{o}s. They also...
Friedlander and Iwaniec proved that the number of points of the orbit $\{\gamma i \colon \gamma\in\SL_2(\Z)\}$ lying at a distance $p-2$ from the origin $i$ of the upper half-plane, with $p\le x$ prime, is of order $x/\log x$; the upper bound is unconditional, while the lower one rests on a strong hypothesis concerning...
Let $E:y^2=x^3-x$. We prove that $\#E(\mathbf{F}_p)/8$ is squarefree and has at most two prime factors for $\gg X/(\log X)^2$ primes $p\le X$. The proof follows the Gaussian CM sieve framework of Iwaniec and Jim\'enez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal $\mat...
Let $q=p^f$, where $p$ is prime. We prove that every $q$-ary linear code meeting the Griesmer bound, with positive dimension and minimum distance $d\ge1$, is $p^{\max\{0,\nu_p(d)-f+1\}}$-divisible, as asserted in Ward's conjecture. Arbitrary projective column multiplicities are allowed. The geometric ingredient is a bo...
Guang-Hui Zhang· 1 citation
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