Related papers: Hilbert cubes in arithmetic sets

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.

Let A and B be finite sets in a commutative group. We bound |A+hB| in terms of |A|, |A+B| and h. We provide a submultiplicative upper bound that improves on the existing bound of Imre Ruzsa by inserting a factor that decreases with h.

Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} +…

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_{\alpha}(T)$ be the upper level sets of Lelong numbers $\nu(T,x)$ of $T$. For any $\alpha\geq \frac{1}{3}$, we show that…

We show that if $\lambda_1,\ldots,\lambda_k$ are algebraic numbers, then $$|A+\lambda_1\cdot A+\dots+\lambda_k\cdot A|\geq H(\lambda_1,\ldots,\lambda_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where…

A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize the definable linear operators on H as exactly the "scalar plus compact" operators.

An s-cap n-flat is given by a set of s points, no three of which are on a common line, in an n-dimensional affine space over the field of three elements. The cap set problem in dimension n is: what is the maximum s such that there is an…

We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.

We study representations of the Cuntz algebras O_d and their associated decompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra UHF_d have an interesting cyclic structure.…

We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic.…

Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This…

Let $X$ be a very general degree $d\geq 5$ hypersurface in $\mathbb{P}^3$. We compute the ample cone of the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ for various small values of $n$ (the answer is already known for large $n$). We obtain…

In this paper, we present several new bounds for the norm and numerical radius of sums of Hilbert space operators. The obtained bounds form a new collection that enriches our understanding of these bounds. We compare our bounds with the…

Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to…

We propose a method for computing upper bounds for the Heilbronn problem for triangles.

In this paper we obtain new lower bounds for the upper box dimension of $\alpha\beta$ sets. As a corollary of our main result, we show that if $\alpha$ is not a Liouville number and $\beta$ is a Liouville number, then the upper box…

In this article we define the $c$-nilpotent multiplier of a finite dimensional Lie suepralgebra. We characterize the structure of $2$-nilpotent multiplier of finite dimensional nilpotent Lie superalgebras whose derived subalgebras have…

Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in…