Thomas Karam

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20222024

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5 papers · 1 filter

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.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,…

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

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

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