An Investigation In Complete Sum-Free Sets

$\require{bbox}\require{ams}$ $$2A=\{a_1+a_2\;|\;a_1,a_2\in A\}$$

Over the course of my spring semester of my first year at Gettysburg College, I begun math research under the supervision of Professor Béla Bajnok using his book Additive Combinatorics: A Menu of Research Problems as my primary source. Within the book, I elected to study an interesting but largely unexplored topic: complete sum-free sets. For anybody unfamilar with this concept, as it is a obscure concept in a relatively obscure field of math, I will provide a brief introduction to this concept and my research below.

We define a sumset of two sets $A$ and $B$, denoted as $A+B$ to be the set of all possible sums picking one element from $A$ and one element from $B$. More precisely, we say that $A+B=\{a+b\;|\;a\in A, b\in B\}$. For example, if we let $A=\{1,3\}$ and $B=\{2,5\}$, then $A+B=\{1+2,1+5,3+2,3+5\}=\{3,5,6,8\}$. Furthermore, we define the $h$-fold sumset of a set $A$, written as $hA$ to be set of all possible sums of $h$ not necessarily distinct elements in $A$. Again for example, the set $A=\{1,3,5\}$ the $2$-fold sumset of $A$ is $2A=\{1+1,1+3,1+5,3+3,3+5,5+5\}=\{2,4,6,8,10\}$.

Now that you know what a sumset is, now I will define what "sum-free" means. Traditionally, a sum-free set usually refers to a set whose $2$-fold sumset has no elements in common with itself. This is written as $2A\cap A=\emptyset$. Note that set $A$ in the previous example is sum-free as $2A$ and $A$ have no elements in common. What I studied is an extended version of the sum-free property, called $(k,l)$-sum-free. A $(k,l)$-sum-free set is a set whose $k$-fold and $l$-fold sumset have no common elements, written as $kA\cap lA=\emptyset$, note that the traditional definition of a sum-free set is a $(2,1)$-sum-free set under the $(k,l)$ definition.

Now is an important time to note, I am studying $(k,l)$-sum-free sets in cyclic groups, meaning instead of simply taking a sum, we will sum within a cyclic group $\mathbb{Z}_n$, and for those of you who are less math savvy, all this just means that after taking our sums, we will just look at the remainder of the sum when divided by $n$, this is called arithmetic modulo (or simply mod) $n$.

Now, to define the final part of my topic of study, completeness. A complete $(k,l)$-sum-free set is a set that in addition to being sum-free in a cyclic group $\mathbb{Z}_n$, if you were to combine the two sets, together, they would form the entire group, written as $kA\cup lA=\mathbb{Z}_n$. Again, for my less math savvy readers, in addition to the two sets $kA$ and $lA$ having no elements in common mod $n$, together, every number between $0$ and $n-1$ appears in one of the two sets. For example if we look at the set $\{1,4\}$ in $\mathbb{Z}_5$, we have $2A=\{1+1,1+4,4+4\}=\{0,2,3\}$. We can see that $A$ and $2A$ have no elements in common, and together, they have every element from $0$ to $4,\;(5-1)$. This means the set $\{1,4\}$ is complete $(2,1)$-sum-free in $\mathbb{Z}_n$

Now, what specifically was I studying? That is a lot harder to answer, but in general, I was investigating (hence the title) the distribution of such sets, and their properties. To give you a better idea of my research, below are a sample of questions related to this research, some of which I have solved, others of which I have not.

Problem 1: For what $n$ does a complete $(k,l)$ sum-free set exist in $\mathbb{Z}_n$?

This is the progress I have achieved thus far with Problem 1. For all $n$ except $1, 3, 7$ and $9$, $\mathbb{Z}_n$ has a complete $(2,1)$-sum-free subset For all $n$ except $1,3,5,9,11,13,15,19,$ and $31$, $\mathbb{Z}_n$ has a complete $(4,1)$-sum-free subset When $k=2$ mod $4$, for all $n\geq k^3+k^2+5k+1$, $\mathbb{Z}_n$ has a complete $(k,1)$-sum-free subset. Another question that I have

Another question that is very interesting is the case of when a complete $(k,l)$-sum-free set is an arithmetic progression, specifically intervals, and this question I have a complete answer to, it is as follows.

An interval $A$ is a complete $(k,l)$-sum-free interval if and only if
(1) $A=\left[\frac{1}{2}\left(\frac{nj}{\gcd(n,k-l)}-\frac{n-2}{k+l}\right),\frac{1}{2}\left(\frac{nj}{\gcd(n,k-l)}+\frac{n-2}{k+l}\right)\right]$
(2) $j$ is odd.
(3) $k+l\;|\;n-2$
(4) $2^B\;|\;n$ where $B$ is the largest non-negative integer such that $2^B\;|\; k-l$
(5) $\frac{nj}{\gcd(n,k-l)}=\frac{n-2}{k+l}$ mod $2$

Quite a complicated, yet complete result (pun intended), but it has very far reaching implications within the scope of complete $(k,l)$-sum-free sets. A follow up question I have to this is the case for when $A$ is the union of two intervals, or the problem

Problem 2: Characterize complete sum-free sets of the form $A=[a,b]\cup[c,d]$

This is the problem I am currently grappling with, but here are some others whose solutions/proofs have evaded me

Problem 3: Is every complete $(2,1)$-sum-free set necessarily symmetric?

Problem 4: Characterize every $n$ for which $\mathbb{Z}_n$ has a complete $(3,1)$-sum-free set.

Problem 5: What is the minimum size of a complete $(2,1)$-sum-free set?

Problem 6: Do any complete $(4,2)$-sum-free sets exist?

This and much more are asked in my full investigation into this topic, which will be posted to this website eventually.