April 15, 2022
In this paper we start to investigate a new body of questions in additive combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+B for subsets of the cyclic group Zp of order p (p prime), and this is just one example of a large family of results. Our aim in this paper is to investigate what happens if we restrict the number of elements of one set that we may use to form the sums. Here is the question we set out to answer: given two subsets, A and B, does B have a subset C of bounded size such that A+C is large, perhaps even comparable to the size of A+B? In particular, can we get close to the lower bound of the Cauchy--Davenport theorem? Our main results show that, rather surprisingly, in many circumstances it is possible to obtain not merely an asymptotic version of the usual sumset bound, but even the exact bound itself.
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June 19, 2022
The classical Cauchy--Davenport inequality gives a lower bound for the size of the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to prove a considerable strengthening of this inequality, where we take only a small number of points from each of the two subsets when forming the sum. One of our results is that there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then there...
October 13, 2022
For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa=|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega(\min(\kappa^{1/3},s)|A|)$. Thus a sumset significantly larger than the Cauchy--Davenport bound can be guaranteed by a...
March 26, 2018
In this paper we present a procedure which allows to transform a subset $A$ of $\mathbb{Z}_{p}$ into a set $ A'$ such that $ |2\hspace{0.15cm}\widehat{} A'|\leq|2\hspace{0.15cm}\widehat{} A | $, where $2\hspace{0.15cm}\widehat{} A$ is defined to be the set $\left\{a+b:a\neq b,\;a,b\in A\right\}$. From this result, we get some lower bounds for $ |2\hspace{0.15cm}\widehat{} A| $. Finally, we give some remarks related to the problem for which sets $A\subset \mathbb{Z}_{p}$ we ha...
March 27, 2023
Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there exist $a_1',a_2'\in A$ such that $a_1+a_2=a_1'+a_2'$ and $\{a_1,a_2\}\neq \{a_1',a_2'\}$. Let $m(p)$ be the size of a smallest subset of $\mathbf{Z}/p\mathbf{Z}$ with no unique sum. The previous best known bounds are $\log p \ll m(p)\ll \sq...
August 2, 2007
It is well known that if S is a subset of the integers mod p, and if the second-largest Fourier coefficient is ``small'' relative to the largest coefficient, then the sumset S+S is much larger than S. We show in the present paper that if instead of having such a large ``spectral gap'' between the largest and second-largest Fourier coefficients, we had it between the kth largest and the (k+1)st largest, the same thing holds true, namely that |S+S| is appreciably larger than |S...
October 27, 2023
This is a survey of old and new problems and results in additive number theory.
April 12, 2013
Let $A$ and $B$ be additive sets of $\mathbb{Z}_{2k}$, where $A$ has cardinality $k$ and $B=v.\complement A$ with $v\in\mathbb{Z}_{2k}^{\times}$. In this note some bounds for the cardinality of $A+B$ are obtained, using four different approaches. We also prove that in a special case the bound is not sharp and we can recover the whole group as a sumset.
November 23, 2020
For $p$ being a large prime number, and $A \subset \mathbb{F}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in $\mathbb{F}_p$, then $|A| < p/8 + o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A = A^*$, then $|A| < p/9 + o(p)$. $(iii)$ If $|A| \gg \frac{\log\log{p}}{\sqrt{\log{p}}}p$, then $|A + A^*| \geqslant (1 - o(1))\min(2\sqrt{|A|p}, p)$. Here the constants $1/8$, $1/9$, and $2$ are the best possible. The proof involves \em...
August 2, 2008
In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\ge min{p,|A|+|B|-1}$, where $A+B={a+b: a\in A, b\in B}$. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. We modify Tao's method so that it ...
July 5, 2024
We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and $|A + A| \leq m$ is at most \[2^{o(s)}\binom{\frac{m+\beta}{2}}{s},\] where $\beta$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ and many other groups. Our result imp...