Related papers: A note on large Kakeya sets

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is…

Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two difficult questions on linear…

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parameterization of this family of knots by a four-tuple of integers, which…

A $(k,\ell )$ partial partition of an $n$-element set is a collection of $\ell $ pairwise disjoint $k$-element subsets. It is proved that, if $n$ is large enough, one can find $\left\lfloor {n\choose k}/{\ell}\right\rfloor$ such partial…

We prove that the largest convex shape that can be placed inside a given convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when "largest" means "largest…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q…

A tope committee K* for a simple oriented matroid M is a subset of its maximal covectors such that every positive halfspace of M contains more than half of the covectors from K*. The structures of the family of all committees for M, and of…

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point…

We produce infinite families of knots $\{K^i\}_{i\geq 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\geq 1}$ is linearly independent in the knot concordance group. We arrange that these examples lie arbitrarily deep in the solvable and…

We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$…

Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound…

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these…

One partially ordered set, $Q$, is a Tukey quotient of another, $P$, denoted $P \geq_T Q$, if there is a map $\phi : P \to Q$ carrying cofinal sets of $P$ to cofinal sets of $Q$. Let $X$ be a space and denote by $\mathcal{K}(X)$ the set of…

We define marked sets and bases over a quasi-stable ideal $\mathfrak j$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded…

We answer a question of Banakh, Jab\l{}o\'nska and Jab\l{}o\'nski by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is…

In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove…

Suppose $\mathcal{X}$ is an $n$-correct set of nodes in the plane, that is, it admits a unisolvent interpolation with bivariate polynomials of total degree less than or equal to $n.$ Then an algebraic curve $q$ of degree $k\le n$ can pass…

In this paper we show a a proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by \(k\)…