Aug 2026· 3 citations· ⚡ 1 influential· 25 references
Mathematics
Abstract
We provide an explicit algebraic construction showing that, uniformly for integers $3 \leq s \leq t$, as $t \to \infty$, \[ R( s,t ) \geq t^{(1-o(1)) \log s / \log(\log s + 1) }. \] For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudl\'ak. In particular, $R(33, t) \geq t^{2.1-o(1)}$, to our knowledge, the first explicit construction showing $R(s, t) \geq t^c$ for some fixed $s$ and some $c>2$. In the diagonal case, it improves the leading constant in the exponent of the classical Frankl--Wilson bound from $1/4$ to $1$, while being almost as simple to describe.
We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. Combined with a previous result of Cohe...
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resol...
Let $\mu$ be the M\"obius function and $e(t)=e^{2\pi it}$. We prove that if $N\ge2$, $\alpha\in\mathbb{R}$, $(a,q)=1$, and $|\alpha-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}\mu^2(n)e(\alpha n)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on...
Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler· 0 citations
For integers $1\le m\le n$, let $s(n,m)$ denote the number of $m$-tuples $(k_1,\ldots,k_m)$ of nonnegative integers satisfying \[ n=\sum_{j=1}^{m}\frac{k_j}{j}. \] We obtain a complete asymptotic expansion for $\log s(n,m)$, uniformly for all $n\ge m$ as $m\to\infty$. The coefficients are given explicitly in terms of l...
For every fixed integer $t \geq 2$, we give an upper bound on the multicolor vector space Ramsey number $R_2(t; k)$ that is a tower function of height independent of $k$. For $t \geq 3$, this is the first bound of its form, significantly improving upon the earlier bounds that are towers of height linear in $k$. We achi...
Let $R_r(k)$ denote the diagonal $r$-colour Ramsey number. We prove that there exist absolute constants $c,K>0$ such that $R_r(k)\le r^{rk}\exp\!\left(-c\frac{k}{r\log^2(2r)}\right)$ for every $r\ge2$ and every $k\ge Kr^2\log^6(2r)$. This improves the exponential saving in a recent bound of Yang and Mao by a factor of...
Sunghyeon Jo· 1 citation
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