Zobrazeno 1 - 10
of 22
pro vyhledávání: '"Karam, Thomas"'
Autor:
Karam, Thomas, Keevash, Peter
We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$ then $E_1,
Externí odkaz:
http://arxiv.org/abs/2411.14145
Autor:
Karam, Thomas
We prove that for every integer $d \ge 2$ there exists a dense collection of subsets of $[n]^d$ such that no two of them have a symmetric difference that may be written as the $d$th power of a union of at most $\lfloor d/2 \rfloor$ intervals. This pr
Externí odkaz:
http://arxiv.org/abs/2408.15144
Autor:
Karam, Thomas
We show that a conceptually simple covering technique has surprisingly rich applications to density theorems and conjectures on patterns in sets involving set differences. These applications fall into three categories: (i) analogues of these statemen
Externí odkaz:
http://arxiv.org/abs/2408.06812
Autor:
Karam, Thomas
We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover another de
Externí odkaz:
http://arxiv.org/abs/2406.10872
Autor:
Karam, Thomas
We identify an assumption on linear forms $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \to \mathbb{F}_p$ that is much weaker than approximate joint equidistribution on the Boolean cube $\{0,1\}^n$ and is in a sense almost as weak as linear independence, bu
Externí odkaz:
http://arxiv.org/abs/2405.04391
Autor:
Karam, Thomas
We provide simple proofs of analogues for coverings numbers of lattices of several recently studied basic statements on the ranks of tensors. We highlight the differences and analogies between the proofs in both settings.
Comment: 15 pages
Comment: 15 pages
Externí odkaz:
http://arxiv.org/abs/2311.03344
Autor:
Karam, Thomas
Let $d \ge 2, h \ge 1$ be integers. Using a fragmentation technique, we characterise $(h+1)$-tuples $(R_1, \dots, R_h, R)$ of non-empty families of partitions of $\{1, \dots, d\}$ such that it suffices for an order-$d$ tensor to have bounded $R_i$-ra
Externí odkaz:
http://arxiv.org/abs/2310.11356
Autor:
Karam, Thomas
Let $(\mathcal{A}_i)_{i \in [s]}$ be a sequence of dense subsets of the Boolean cube $\{0,1\}^n$ and let $p$ be a prime. We show that if $s$ is assumed to be superpolynomial in $n$ then we can find distinct $i,j$ such that the two distributions of ev
Externí odkaz:
http://arxiv.org/abs/2309.14229
Autor:
Karam, Thomas
Let $d \ge 3$ be an integer. We show that whenever an order-$d$ tensor admits $d+1$ decompositions according to Tao's slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordi
Externí odkaz:
http://arxiv.org/abs/2308.07101
Autor:
Gowers, W. T., Karam, Thomas
Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0<\epsilon\leq 1$. We show that there exists a constant $C=C(p, \epsilon)$ such that for every positive integer $k$, whenever $\phi_1, \dots, \phi_k: \mathbb{F}_p^n \rightar
Externí odkaz:
http://arxiv.org/abs/2306.00747