Thomas Karam

activity

20222024

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6 papers · 1 filter

math.CO2023

Simple proofs for lattice coverings and sparse tensors

Thomas Karam

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 analo…

math.CO2023

The interplay between bounded ranks of tensors arising from partitions

Thomas Karam

Let $d \geq 2, h \geq 1$ be integers. Using a fragmentation technique, we characterise $(h + 1)$ -tuples $(R_{1}, \dots, R_{h}, R)$ of non-empty families of partitions of ${1, \dots, d}$ …

math.CO2023

Small sunflowers and the structure of slice rank decompositions

Thomas Karam

Let $d \geq 3$ 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-varia…

math.CO2023

Low-complexity approximations for sets defined by generalizations of affine conditions

W. T. Gowers, Thomas Karam

Let $p$ be a prime, let $S$ be a non-empty subset of $F_{p}$ and let $0 < ε \leq 1$ . We show that there exists a constant $C = C (p, ε)$ such that for every positive integer $k$ ,…

math.CO2023

Ranges of polynomials control degree ranks of Green and Tao over finite prime fields

Thomas Karam

Let $p$ be a prime, let $1 \leq t < d < p$ be integers, and let $S$ be a non-empty subset of $F_{p}$ . We establish that if a polynomial $P : F_{p}^{n} \to F_{p}$ wi…

math.CO2023

On subtensors of high partition rank

Jan Draisma, Thomas Karam

We prove that for every positive integer $d \geq 2$ there exist polynomial functions $F_{d}, G_{d} : N \to N$ such that for each positive integer $r$ , every order- $d$ t…