Related papers: Marstrand type slicing statements in $\mathbb{Z}^{…

Borsuk's conjecture states that any bounded set in R^n can be partitioned into n+1 sets of smaller diameter. It is known to be false for all n bigger or equal to 323. Here we show that Borsuk's conjecture fails in dimensions 321 and 322.…

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete…

Francisco Santos has described a new construction, per- turbing apart a non-simple face, to offer a counterexample to the Hirsch Conjecture. We offer two observations about this perturbed wedge con- struction, regarding its effect on…

We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d=2 (the original…

We generalize the ham sandwich theorem for the case of well separated measures. Given convex bodies $K_1,...,K_d$ in $\mathbb{R_d}$ and numbers $\alpha_1,...,\alpha_d \in [0, 1]$, we give a sufficient condition for existence and uniqueness…

By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We provide a new…

This paper deals with the reconstruction of a discrete measure $\gamma_Z$ on $\mathbb{R}^d$ from the knowledge of its pushforward measures $P_i\#\gamma_Z$ by linear applications $P_i: \mathbb{R}^d \rightarrow \mathbb{R}^{d_i}$ (for instance…

We show that the hyperplane conjecture holds for the classes of $k$-intersection bodies with arbitrary measures in place of volume.

Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding…

A new dimension reduction method based on Gaussian finite mixtures is proposed as an extension to sliced inverse regression (SIR). The model-based SIR (MSIR) approach allows the main limitation of SIR to be overcome, i.e., failure in the…

A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where…

Let R=K[M] be a normal affine monoid algbera over a field K.Up to isomorphism the conic ideals are exactly the direct summands ofthe extension R^{1/n} of R. We show that the classes of the conic divisorial ideals can be identified with the…

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(\delta,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at…

We study the Hausdorff and box-counting dimensions of cookie-cutter-like sets formed by sequential dynamics of a finite number of expanding maps. Under some natural conditions, these dimensions turn out to be the minimum and maximum of the…

Transposed Poisson structures on the Schr\"{o}dinger algebra in $(n+1)$-dimensional space-time of Schr\"{o}dinger Lie groups are described. It was proven that the Schr\"{o}dinger algebra $\mathcal{S}_{n}$ in case of $n\neq 2$ does not have…

It was shown by Bishop that if Thurston's $K = 2$ conjecture holds for some planar domain, then Brennan's conjecture holds for the Riemann map of that domain as well. In this paper we show numerically that the original counterexample to…

In this second part of the paper, dedicated to theories with extra dimensions, a new physical notion about the "tensor length scale" is introduced, based on the gravitational theories with covariant and contravariant metric tensor…

Let X be a smooth projective variety over an algebraically closed field of positive characteristic. We prove that if D is a pseudo-effective R-divisor on X which is not numerically equivalent to the negative part in its divisorial Zariski…

We show that, also within the class of representation-tame finite dimensional algebras $\Lambda$, the big left finitistic dimension of $\Lambda$ may be strictly larger than the little. In fact, the discrepancies $Findim \Lambda - findim…

The results in this paper are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. In particular, given any finite collection of…