Related papers: The Correct Exponent for the Gotsman-Linial Conjec…

Let $I(n,l)$ denote the maximum possible number of incidences between $n$ points and $l$ lines. It is well known that $I(n,l) = \Theta(n^{2/3}l^{2/3} + n + l)$. Let $c_{\mathrm{SzTr}}$ denote the lower bound on the constant of…

In the $d$-Scattered Set problem we are asked to select at least $k$ vertices of a given graph, so that the distance between any pair is at least $d$. We study the problem's (in-)approximability and offer improvements and extensions of…

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex…

The angular resolution of a planar straight-line drawing of a graph is the smallest angle formed by two edges incident to the same vertex. Garg and Tamassia (ESA '94) constructed a family of planar graphs with maximum degree $d$ that have…

Gaussian universality results assert that the properties of many estimators remain unchanged when the input data are replaced by Gaussians. Such results have gained popularity in high-dimensional statistics and machine learning, as…

We show that a Boolean degree $d$ function on the slice $\binom{[n]}{k}$ is a junta if $k \geq 2d$, and that this bound is sharp. We prove a similar result for $A$-valued degree $d$ functions for arbitrary finite $A$, and for functions on…

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…

Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log…

The Bateman--Horn Conjecture predicts how often an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes prime values. We demonstrate that with sufficient averaging in the coefficients of $f$ (viz. exponential in the size of the inputs),…

Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with…

We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$. This problem is conjectured to have a sharp feasibility transition: for…

We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{n\pi^2}{12\log2}$. The exponential rate is best possible,…

The \emph{signed series} problem in the $\ell_2$ norm asks, given set of vectors $v_1,\ldots,v_n\in \mathbf{R}^d$ having at most unit $\ell_2$ norm, does there always exist a series $(\varepsilon_i)_{i\in [n]}$ of $\pm 1$ signs such that…

We use the Petrow-Young [10] subconvexity bound for Dirichlet $L$-functions to show that $d_4(n)$ has exponent of distribution $4/7$ when we allow an average over $a$ mod $q$, thereby giving an equidistribution result for $d_4(n)$ which…

We consider tensor product random fields $Y_d$, $d\in\mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…