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Oriol Serra

Publications and source records attributed to Oriol Serra.

47 records · Page 3Linked to original sources
arXiv2011

Counting patterns in colored orthogonal arrays

Let $S$ be an orthogonal array $OA(d,k)$ and let $c$ be an $r$--coloring of its ground set $X$. We give a combinatorial identity which relates the number of vectors in $S$ with given color patterns under $c$ with the cardinalities of the color classes. Several applications of the identity are considered. Among them, we show that every equitable $r$--coloring of the integer interval $[1,n]$ has at least $1/2(n/r)^2+O(n)$ monochromatic Schur triples. We also show that in an orthogonal array $OA(d,d-1)$, the number of monochromatic vectors of each color depends only on the number of vectors which miss that color and the cardinality of the color class.

math.CO↗
arXiv2011

Rainbow-free 3-colorings of Abelian Groups

A 3-coloring of the elements of an abelian group is said to be rainbow--free if there is no 3-term arithmetic progression with its members having pairwise distinct colors. We give a structural characterization of rainbow--free colorings of abelian groups. This characterization proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow-free 3-coloring of cyclic groups.

math.CO↗
arXiv2010

On the cardinality of sumsets in torsion-free groups

Let $A, B$ be finite subsets of a torsion-free group $G$. We prove that for every positive integer $k$ there is a $c(k)$ such that if $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0k^{6}$ for arbitrary torsion-free groups, and $c(k)<c_0k^{3}$ for groups with the unique product property, where $c_0$ is an absolute constant. We give examples to show that $c(k)$ is at least quadratic in $k$.

math.GR↗
arXiv2008

A Removal Lemma for Systems of Linear Equations over Finite Fields

We prove a removal lemma for systems of linear equations over finite fields: let $X_1,...,X_m$ be subsets of the finite field $\F_q$ and let $A$ be a $(k\times m)$ matrix with coefficients in $\F_q$ and rank $k$; if the linear system $Ax=b$ has $o(q^{m-k})$ solutions with $x_i\in X_i$, then we can destroy all these solutions by deleting $o(q)$ elements from each $X_i$. This extends a result of Green [Geometric and Functional Analysis 15(2) (2005), 340--376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.

math.CO↗
arXiv2008

A combinatorial proof of the Removal Lemma for Groups

Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

math.CO↗
arXiv2008

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

We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < 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.

math.NT↗
arXiv2007

Properties of two dimensional sets with small sumset

Let $A, B\subseteq \mathbb{R}^2$ be finite, nonempty subsets, let $s\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines, $\ell_1,...,\ell_{t},\ell'_1,...,\ell'_{t}$, with $A\subseteq \bigcup_{i=1}^{t}\ell_i$ and $B\subseteq\bigcup_{i=1}^{t}\ell'_i$. Suppose $h_1(A,B)\geq s$. Then we show that: (a) if $||A|-|B||\leq s$ and $|A|+|B|\geq 4s^2-6s+3$, then $$|A+B|\geq (2-\frac 1 s)(|A|+|B|)-2s+1;$$ (b) if $|A|\geq |B|+s$ and $|B|\geq 2s^2-{7/2}s+{3/2}$, then $$|A+B|\geq |A|+(3-\frac 2 s)|B|-s;$$ (c) if $|A|\geq {1/2}s(s-1)|B|+s$ and either $|A|> {1/8}(2s-1)^2|B|-{1/4}(2s-1)+\frac{(s-1)^2}{2(|B|-2)}$ or $|B|\geq \frac{2s+4}{3}$, then $$|A+B|\geq |A|+s(|B|-1).$$ This extends the 2-dimensional case of the Freiman $2^d$--Theorem to distinct sets $A$ and $B$, and, in the symmetric case $A=B$, improves the best prior known bound for $|A|+|B|$ (due to Stanchescu, and which was cubic in $s$) to an exact value. As part of the proof, we give general lower bounds for two dimensional subsets that improve the 2-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, and that generalize the 2-dimensional case of the Brunn-Minkowski Theorem.

math.CO↗
arXiv2007

Connectivity of Addition Cayley Graphs

For any finite abelian group $G$ and any subset $S\seq G$, we determine the connectivity of the addition Cayley graph induced by $S$ on $G$. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a special, explicitly described form.

math.CO↗
arXiv2007

On Some Subgroup Chains Related to Kneser's Theorem

A recent result of Balandraud shows that for every subset S of an abelian group G, there exists a non trivial subgroup H such that |TS| <= |T|+|S|-2 holds only if the stabilizer of TS contains H. Notice that Kneser's Theorem says only that the stabilizer of TS must be a non-zero subgroup. This strong form of Kneser's theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud's results in the abelian case.

math.NT↗
arXiv2005

On the critical pair theory in Z/pZ

Let A and B be subsets of Z/pZ such that |A+B| < |A|+|B|+2. We prove that, if |A|>3, |B|>4, |A+B| 52, then A and B are included in arithmetic progressions with the same difference and of size |A|+2 and |B|+2 respectively. This extends the well-known theorem of Vosper and a recent result of Rodseth and one of the present authors.

math.NT↗