Welcome to MathOverflow

Question

1

I am an algebraist, and I am wondering if there is a definition for the following:

Let $A_1$, $A_2$, $\ldots, A_n$ be sets of integers (or more generally, subsets of a group $G$). Say that (for the purposes of this question) $A_1\times A_2\cdots\times A_n$ is special provided that whenever $a_1+a_2+\cdots+a_n=b_1+b_2\cdots+b_n$ with each $a_i,b_i\in A_i$, $1\leq i\leq n$, then $a_i=b_i$ for all $i$, $1\leq i\leq n$.

Is there some terminology for this property? Any information would be appreciated.

edited Dec 20 at 8:08

Martin Brandenburg
29.1k●2●45●129

asked Dec 20 at 1:48

Greg
11●1

In the case that $A_1,\dotsc,A_n$ are singletons, this notion as known as unique-sum sets (see e.g. cmc11.uni-jena.de/proceedings/frisco.pdf). – Martin Brandenburg Dec 20 at 8:07

@Martin: I suspect there must be something wrong in your statement because if $A_1, \ldots, A_n$ are singletons, then also $A_1 \times \ldots \times A_n$ is a singleton, and hence the condition in the question is trivially true. – boumol Dec 20 at 8:26

I think Martin is referrring to $B_n[1]$ sets which would -- I think! -- be the case $A_1=A_2=\dots= A_n$ – Yemon Choi Dec 20 at 9:37

While what Yemon Choi says is also a pertinent related notion Martin Brandenburg meant something else. Namely, two elements sets (not singletons). The point is that one studies the question when a set (or also multiset) has the property that all the sums of elements of distinct subsets are actually distinct. This corresponds directly to the case that the A_i are of the form {0,a_i} in the present problem. Yet since the present problem is invarinat under shifts this also applies in case each A_i contains (at most) two elements. – quid Dec 20 at 11:30

1

A more natural way to state the condition is that $|A_1+\cdots+A_h| = |A_1| \cdots |A_h|$ (the parameter $h$ is more common than $n$ here). These come up in additive combinatorics, but I don't know a name for them. If $A_1= \dots =A_h$, and you don't care about the ordering of the sum (i.e., one can reorder the $b_i$ so that $a_i=b_i$), these are called "Sidon Sets", also $B_h$-sets. – Kevin O'Bryant Dec 20 at 19:30

Aaron Meyerowitz · Answer 1 · 2012-12-20 11:11:21Z UTC

There is a lot of interesting work using the term "Tiling" (sometimes " Algebraic Tiling"), especially when there are two factors although all the factors but one can be combined (I am thinking mainly of Abelian Groups). If $A$ is any infinite set of integers which increases quickly enough then there is a $B$ with $A \oplus B= \mathbb{Z}.$ For $A \oplus B=\mathbb{Z}$ with $A$ finite, there are results and open questions, cyclotomic polynomials are a useful tool For $A \oplus B= \mathbb{N}$ (bases for the positive integers) both $A$ and $B$ are highly structured. Evidently there are connections to Musical Canons.

There have uses of non-abelian factorizations for cryptography.

There are better and other references but that gives a few entry points

unique sums in a finite direct product of sets of integers

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.