Signed Zero-Sum-Free Sets and Signed Sum-Free Sets

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

This summer, one topic that I have tackled is the concept of signed zero-sum-free sets. Currently, there is nowhere near enough content to make a paper out of, thus I decided to make a blog post on the topic.

We define a h-fold signed sumset of a set $A$, written as $h_± A$ to be the set of all possible combinations of adding and subtracting $h$ (not nessicarily distinct) elements together from $A$, with the only rule being that you cannot add and subtract the same element when determining a term in $h_± A$. For example, given the set $A=\{a,b,c\}$: $a+a-b\in 3_± A$ and $-a-b+c\in 3_±A$ but, $a+b-b∉ 3_±A$.

The next step is to define a zero-$h$-signed sum-free set. A set $A$ meets this criteria if and only if $0∉h_±A$. In this blog post, we will be discussing this concept only in regards to finite abelian groups, chiefly cyclic groups.

The main question I am seeking to answer is as follows

Problem 1: Find a formula for $\tau_±(\mathbb{Z}_n,h)$, where $\tau_±(\mathbb{Z}_n,h)$ is equal to the maximum cardinality of a zero-$h$-signed sum-free subset of $\mathbb{Z}_n$.

This is the progress I have achieved thus far with Problem 1.

It is important to note that it is also possible to easily arrive at the first result with the impressive results of Francis, Slevin, and Szanto which can be found alongside a paper of mine in the 23rd edition of Béla Bajnok's locally published Research Papers in Mathematics.

$\tau_±(\mathbb{Z}_n,3)=v_3(n,3)$ where $v_g(n,h)=\max\left\{\frac{n}{d}\left\lfloor\frac{d-1-\gcd(d,g)}{h}+1\right\rfloor\;|\;d\in D(n)\right\}$ Where $D(n)$ is the set of positive divisors of $n$.

I was able to derive this result from an unpblished result of mine, which I will share in the future once it finishes the peer review process as I hope to get this result published. This result concerns the same question we are asking today, but for the standard sumset case, and is much more general than the results being presented here.

The case for $\tau_±(\mathbb{Z}_n,h)$ for even $h$ is much more complex, however I do have the followong results in regards to $\tau_±(\mathbb{Z}_n,5)$ For prime $p\geq 7$, we have that if $p=1,2,$ or $4$ mod $5$ then $\tau_±(\mathbb{Z}_p,5)=\left\lfloor\frac{p+3}{5}\right\rfloor$ if $p=3$ mod $5$ then $\tau_±(\mathbb{Z}_p,5)=\frac{p-3}{5}$. Furthermore, if $n$ is even $\tau_±(\mathbb{Z}_n,5)=\frac{n}{2}$, and lastly if $n$ is divisible by 5 but not any numbers 2, 3, or 4 mod 5 then $\tau_±(\mathbb{Z}_n,5)=\frac{n}{5}-1$

This leaves one last case, and that is the remaining odd composite values of $n$. These are much trickier to develop a proof for than the prime case.

I have done very little work in regards to larger values of $h$, however I do have that $\tau_±(\mathbb{Z}_p,7)=\left\lfloor\frac{p+5}{7}\right\rfloor$ if $p$ is $1,2,4,$ or $6$ mod $7$. I believe that for the remaining cases, $\tau_±(\mathbb{Z}_p,7)=\left\lfloor\frac{p+5}{7}\right\rfloor-1$.

In a similar vien to to above topic, I would also like to discuss signed-$(k,l)$-sum-free sets, and weak signed-$(k,l)$-sum-free sets. Before I define these two ideas, I will first define a restricted signed sumset. the $h$-fold restricted signed sumset of a set $A$, written as $h\hat{_±}A$, is the set of all distinct combinations of $h$ elements in $A$ added and subtracted together. $A$ is signed-$(k,l)$-sum-free set if and only of $k_±A\cap l_±A=\emptyset$, and weak signed-$(k,l)$-sum-free if and only if $k\hat{_±}A\cap l\hat{_±}A=\emptyset$.

There are three results regarding this topic that I would like to share, but first I must introduce a notation in the same vien as $\tau_±$.

$\mu_±(\mathbb{Z}_n,\{k,l\})$ is equal to the cardinality of the largest signed-$(k,l)$-sum-free subset of $\mathbb{Z}_n$. $\mu\hat{_±}(\mathbb{Z}_n,\{k,l\})$ is equal to the cardinality of the largest weak signed-$(k,l)$-sum-free subset of $\mathbb{Z}_n$. The results are as follows $\mu_±(\mathbb{Z}_n,\{2,1\})=v_3(n,3)$ $\mu\hat{_±}(\mathbb{Z}_n,\{2,1\})=v_1(n,3)$ For all cases classified in Theorem 3, $\mu_±(\mathbb{Z}_n,\{4,1\})=\tau_±(\mathbb{Z}_n,5)$

Theorems 6 and 8 certianly imply a connection between $\mu_±$ and $\tau_±$, and that is no coincidence. For a set to be signed-$(k,l)$-sum-free, it must be zero-$k+l$-signed sum-free, among other things. In fact, its not hard to establish (despite its complexity that) $\mu_±(\mathbb{Z}_n,\{3,1\})=\tau_±(\mathbb{Z}_n,4)$. So, I conjecture that for $n>k+l$ $\mu_±(\mathbb{Z}_n,\{k,l\})=\tau_±(\mathbb{Z}_n,k+l)$

To conclude this post, I will pose several questions regarding this topic

Problem 2: Find $\tau_±(\mathbb{Z}_n,5)$ for odd composite values of $n$.

Problem 3: Find $\tau_±(\mathbb{Z}_n,4)$ for an infinite subset of $n$.

Problem 4: Find $\tau_±(\mathbb{Z}_p,7)$ for prime $p$ congruent to 3 and 5 mod 7.

Problem 5: Find $\mu_±(\mathbb{Z}_n,\{k,l\})$ for $(k,l)\neq(2,1)$

Problem 6: Find $\mu\hat{_±}(\mathbb{Z}_n,\{k,l\})$ for $(k,l)\neq(2,1)$

Problem 7: Establish equality between $\mu_±(\mathbb{Z}_n,\{k,l\})$ and $\tau_±(\mathbb{Z}_n,k+l)$, or find cases which equality does not hold.

Problem 8: Explore $\tau_±(G,h)$, $\mu_±(G\{k,l\})$, and $\mu\hat{_±}(G,\{k,l\})$for non-cyclic, abelian $G$.