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### A characterization of sequences with the minimum number of k-sums modulo k

Colloquium Mathematicae

Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.

### A note on minimal zero-sum sequences over ℤ

Acta Arithmetica

A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a₁,...,{a}_{h}$ and negative terms $b₁,...,{b}_{k}$. We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $\sigma ⁺={\sum }_{i=1}^{h}{a}_{i}=-{\sum }_{j=1}^{k}{b}_{j}$. These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive integer n.

### A note on sumsets of subgroups in $ℤ{*}_{p}$

Acta Arithmetica

Let A be a multiplicative subgroup of $ℤ{*}_{p}$. Define the k-fold sumset of A to be $kA={x}_{1}+...+{x}_{k}:{x}_{i}\in A,1\le i\le k$. We show that $6A\supseteq ℤ{*}_{p}$ for $|A|>{p}^{11/23+ϵ}$. In addition, we extend a result of Shkredov to show that ${|2A|\gg |A|}^{8/5-ϵ}$ for $|A|\ll {p}^{5/9}$.

### A spectral gap theorem in SU$\left(d\right)$

Journal of the European Mathematical Society

We establish the spectral gap property for dense subgroups of SU$\left(d\right)$$\left(d\ge 2\right)$, generated by finitely many elements with algebraic entries; this result was announced...

Acta Arithmetica

### Capturing forms in dense subsets of finite fields

Acta Arithmetica

An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field ${}_{q}$. Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size of $A{\subset }_{q}$ to guarantee...

Acta Arithmetica

Acta Arithmetica

### Expansion in $S{L}_{d}\left({𝒪}_{K}/I\right)$, $I$ square-free

Journal of the European Mathematical Society

Let $S$ be a fixed symmetric finite subset of $S{L}_{d}\left({𝒪}_{K}\right)$ that generates a Zariski dense subgroup of $S{L}_{d}\left({𝒪}_{K}\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_{d}\left({𝒪}_{K}/I\right)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_{K}$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=ℚ$.

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### Large sets with small doubling modulo $p$ are well covered by an arithmetic progression

Annales de l’institut Fourier

We prove that there is a small but fixed positive integer $ϵ$ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $|2S|\le \left(2+ϵ\right)|S|$ and $2\left(|2S|\right)-2|S|+3\le p$ is contained in an arithmetic progression of length $|2S|-|S|+1$. This is the first result of this nature which places no unnecessary restrictions on the size of $S$.

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

Acta Arithmetica

### On the Davenport constant and group algebras

Colloquium Mathematicae

For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S=g₁·...·{g}_{l}$ over G such that $\left({X}^{g₁}-a₁\right)·...·\left({X}^{{g}_{l}}-{a}_{l}\right)\ne 0\in K\left[G\right]$ for all $a₁,...,{a}_{l}\in {K}^{×}$. If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which (G) -...

### On the dimension of additive sets

Acta Arithmetica

We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between these dimensions by improving an inequality of Lev and Yuster, and we show that these bounds are asymptotically sharp, using in particular the existence of large dissociated subsets of {0,1}ⁿ ⊂ ℤⁿ.

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