Related papers: A Sauer-Shelah-Perles Lemma for Sumsets

The multiplicity (resp. degree) of a function $f$ relative to a semianalytic subset $S$ of $\mathbb{R}^n$ is the greatest (resp. smallest) exponent among numbers $j$ such that the inequality $|f(x)|\leq C\|x\|^j$ holds on $S$ near $0$…

Let $S$ be a set of $n$ points in general position in the plane. The Second Selection Lemma states that for any family of $\Theta(n^3)$ triangles spanned by $S$, there exists a point of the plane that lies in a constant fraction of them.…

For a sequence $S$ of terms from an abelian group $G$ of length $|S|$, let $\Sigma_n(S)$ denote the set of all elements that can be represented as the sum of terms in some $n$-term subsequence of $S$. When the subsum set is very small,…

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C…

We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.

Let $\mathcal{A}$ be a union-closed family of sets with universe $\bigcup_{A \in \mathcal{A}}A = [n] = \{1,\cdots,n\}$ and length $\ell$. We prove that $|\mathcal{A}| \leq \sum_{i=0}^{\ell} \binom{n}{i}$, with equality if and only if…

A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in…

Let $A, B\subseteq \mathbb{R}^2$ be finite, nonempty subsets, let $s\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines,…

It is shown that the $L^\alpha$-norms polynomials Rudin conjecture fails. Our counterexample is inspired by Bourgain's work on NLS. Precisely, his study of the Strichartz's inequality of the $L^6$-norm of the periodic solutions given by the…

How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight…

Let $n$ be a large number. A subset $A$ of $Z_n$ is complete if $S_A = Z_n$, where $S_A$ is the collection of the subset sums of $A$. Olson proved that if $n$ is a prime and $|A|> 2n^{1/2} $, then $S_A$ is complete. We show that a similar…

Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $\sigma_1, \ldots, \sigma_H \in \mathfrak{S}_n$, there exist finite subsets $A_1,…

Let $r \geq 2$, $n$ and $k$ be integers satisfying $k \leq \frac{r-1}{r}n$. In the original arXiv version of this note we suggested a conjecture that the family of all $k$-subsets of an $n$-set cannot be partitioned into fewer than $\lceil…

Let A, B and S be three subsets of a finite Abelian group G. The restricted sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A, b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y in…

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes,…

In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$ is irreducible. This conjecture remains open, but it…

We prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^(r) are adjacent if |x \Delta y|=2. Namely, if \mathcal{A} \subset [n]^(r) with |\mathcal{A}|=\alpha {n \choose r},…

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(\omega, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can…

We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit…

Dvir and Moran proved the following upper bound for the size of a family $\mbox{$\cal F$}$ of subsets of $[n]$ with $\mbox{Vdim}(\mbox{$\cal F$} \Delta \mbox{$\cal F$})\leq d$. Let $d\leq n$ be integers. Let $\mbox{$\cal F$}$ be a family of…