Related papers: On uniform polynomial approximation

We say that $d_3(n)$ has exponent of distribution $\theta$ if, for every $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved, following earlier work of Banks, Heath-Brown,…

Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce…

We present a new exponential inequality as a generalization of that of Sung \textit{et al.} \cite{sun2011} for $M$-acceptable random variables, and hence for extended negative ones. Our result is based on the simple real inequality $e^{x}…

In the Random Subset Sum Problem, given $n$ i.i.d. random variables $X_1, ..., X_n$, we wish to approximate any point $z \in [-1,1]$ as the sum of a suitable subset $X_{i_1(z)}, ..., X_{i_s(z)}$ of them, up to error $\varepsilon$. Despite…

Motivated by questions in number theory, Myerson asked how small the sum of 5 complex nth roots of unity can be. We obtain a uniform bound of O(n^{-4/3}) by perturbing the vertices of a regular pentagon, improving to O(n^{-7/3}) infinitely…

We prove that there is an absolute constant $ C$ such that for every $ n \geq 2 $ and $ N\geq 10^n, $ there exists a polytope $ P_{n,N} \subset \mathbb{R}^n $ with at most $ N $ facets that satisfies…

We consider the probability that the random signed sum $\xi_1 v_1 + \dotsb + \xi_n v_n$ lies within a given distance $r$ of the origin, where $v_1,\dotsc,v_n \in \mathbb{R}^d$ are fixed unit vectors and $\xi_1,\dotsc,\xi_n$ are…

In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman's Question, which asked for a bound on the projective dimension of a graded polynomial ideal purely in terms of the number and degrees of its generators.…

Let $\theta >2$ be real and non-integral with integer part $n = \lfloor \theta \rfloor$ and let $ \phi (x)$ be a generalised polynomial with leading term $x^\theta.$ We establish a mean value estimate for the exponential sum…

Let $f_1(x),\ldots,f_n(x)$ be some polynomials. The upper bound on the number of $x\in\mathbb F_p$ such that $f_1(x),\ldots,f_n(x)$ are roots of unit of order $t$ is obtained. This bound generalize the bound of the paper \cite{V-S} to the…

We give a sharp lower bound on the capacity of a real stable polynomial, depending only on the value of its gradient at $x = 1$. This result implies a sharp improvement to a similar inequality proved by Linial-Samorodnitsky-Wigderson in…

We study the number $N(n, A_n, X)$ of number fields of degree $n$ whose Galois closure has Galois group $A_n$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(n, A_n, X) \sim C_n X^{1/2} (\log X)^{b_n}$,…

In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…

In this paper, we shed new light on the spectrum of the relation algebra we call $A_{n}$, which is obtained by splitting the non-flexible diversity atom of $6_{7}$ into $n$ symmetric atoms. Precisely, we show that the minimum value in…

For every $p\leq n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-\big(2\binom{2p}{p+1}-\binom{2p-2}{p-1}+2\big)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one…

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega=(\omega_n)_{n\geq 1}$ uniformly distributed…

We consider polynomials $p_n^{\omega}(x)$ that are orthogonal with respect to the oscillatory weight $w(x)=e^{i\omega x}$ on $[-1,1]$, where $\omega>0$ is a real parameter. A first analysis of $p_n^{\omega}(x)$ for large values of $\omega$…

The main contribution of this paper is a new improved variant of the laser method for designing matrix multiplication algorithms. Building upon the recent techniques of [Duan, Wu, Zhou, FOCS 2023], the new method introduces several new…

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. Approximate degree is known to be a lower bound on quantum query complexity. We resolve or nearly…