Positive configuration space

Polynomial configurations in sets of positive upper density over local fields

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of $\mathbb{R}^m$ with respect to the portion of the graph of $F$ defined by $a\leq \log |s| \leq T$ is at most $O(1/(T-a))$. We conclude that if $I \subseteq \mathbb{R}^m$ has positive upper density, then the difference set $I-I$ contains vectors of the form $F(s)$ f…

Congruence classes of large configurations in vector spaces over finite fields

Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such that any set $E\subset \mathbb{F}_q^d$ with $|E|\gtrsim q^s$ determines $\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations for $k\leq d$. Because congruence classes can be identified with tuples of distances between distinct points when $k\leq d$, and because there are $\binom{k+1}{2}$ such pairs, this means any such $E$ determines a positive proportion of all congruence classes. In the $k>d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{\binom{k+1}{2}}$ is…

欧几里得空间和整数格的稠密子集里的单形中心构型

Simplex--center configurations in dense subsets of Euclidean spaces and the integer lattice

该研究针对欧几里得空间及整数格的稠密子集，证明了含单形及其重心的构型的密度拉姆齐定理，在特定条件下推广了相关度量下的等距副本存在性结果。

Distance Graphs and sets of positive upper density in $\mathbb{R}^d$

We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance graphs.

Configuration spaces of $\mathbb{C} \setminus k$

In this note, we collect mostly known formulas and methods to compute the standard and virtual Poincaré polynomials of the configuration spaces of the plane $\mathbb{C} \setminus k$ with $k$ deleted points and compare the answers.

正密度集合中的无限线性模式

Infinite linear patterns in sets of positive density

本文研究正密度集合中无限线性模式的性质，推广了Szemerédi定理和密度有限和定理。

Bounds on the number of 2-level polytopes, cones and configurations

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones and d-configurations to faces of the correlation cone. We complement this with a $2^{Ω(d^2)}$ lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite…

Models for Configuration Space in a Simplicial Complex

We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve $y^2 z = x^3 + x^2 z$, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its tw…

Orders of the canonical vector bundles over configuration spaces of finite graphs

We prove that the order of the canonical vector bundle over the configuration space is $2$ for a general planar graph, and is $4$ for a nonplanar graph.