Thomas Karam

activity

20222024

most citedHigh-rank subtensors of high-rank tensors

1 citations · 1 across the 9 of their papers we have counts for

collaborators

10 papers

math.CO2024

Unions of intervals in codes based on powers of sets

Thomas Karam

We prove that for every integer $d \geq 2$ 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 $d$ t…

math.CO2024

Three applications of coverings to difference patterns

Thomas Karam

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 ap…

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…

math.CO2022

Equidistribution of high-rank polynomials with variables restricted to subsets of $F_{p}$

W. T. Gowers, Thomas Karam

Let $p$ be a prime and let $S$ be a non-empty subset of $F_{p}$ . Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields,…

math.CO2022★ 1 cited

High-rank subtensors of high-rank tensors

Thomas Karam

Let $d \geq 2$ be an integer. We show that for a class of notions $R$ of rank for order- $d$ tensors, which includes in particular the tensor rank, the slice rank and the partition r…

Unions of intervals in codes based on powers of sets

Three applications of coverings to difference patterns

Simple proofs for lattice coverings and sparse tensors

The interplay between bounded ranks of tensors arising from partitions

Small sunflowers and the structure of slice rank decompositions

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

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

On subtensors of high partition rank

Equidistribution of high-rank polynomials with variables restricted to subsets of Fp​

High-rank subtensors of high-rank tensors

Equidistribution of high-rank polynomials with variables restricted to subsets of $F_{p}$