I’ve uploaded a new paper to the arXiv entitled “The sum-product phenomenon in arbitrary rings“, and submitted to Contributions to Discrete Mathematics. The sum-product phenomenon asserts, very roughly speaking, that given a finite non-empty set A in a ring R, then either the sum set or the product set will be significantly larger than A, unless A is somehow very close to being a subring of R, or if A is highly degenerate (for instance, containing a lot of zero divisors). For instance, in the case of the integers , which has no non-trivial finite subrings, a long-standing conjecture of Erdös and Szemerédi asserts that for every finite non-empty and every . (The current best result on this problem is a very recent result of Solymosi, who shows that the conjecture holds for any greater than 2/3.) In recent years, many other special rings have been studied intensively, most notably finite fields and cyclic groups, but also the complex numbers, quaternions, and other division algebras, and continuous counterparts in which A is now (for instance) a collection of intervals on the real line. I will not try to summarise all the work on sum-product estimates and their applications (which range from number theory to graph theory to ergodic theory to computer science) here, but I discuss this in the introduction to my paper, which has over 50 references to this literature (and I am probably still missing out on a few).
I was recently asked the question as to what could be said about the sum-product phenomenon in an arbitrary ring R, which need not be commutative or contain a multiplicative identity. Once one makes some assumptions to avoid the degenerate case when A (or related sets, such as A-A) are full of zero-divisors, it turns out that there is in fact quite a bit one can say, using only elementary methods from additive combinatorics (in particular, the Plünnecke-Ruzsa sum set theory). Roughly speaking, the main results of the paper assert that in an arbitrary ring, a set A which is non-degenerate and has small sum set and product set must be mostly contained inside a subring of R of comparable size to A, or a dilate of such a subring, though in the absence of invertible elements one sometimes has to enlarge the ambient ring R slightly before one can find the subring. At the end of the paper I specialise these results to specific rings, such as division algebras or products of division algebras, cyclic groups, or finite-dimensional algebras over fields. Generally speaking, the methods here give very good results when the set of zero divisors is sparse and easily describable, but poor results otherwise. (In particular, the sum-product theory in cyclic groups, as worked out by Bourgain and coauthors, is only recovered for groups which are the product of a bounded number of primes; the case of cyclic groups whose order has many factors seems to require a more multi-scale analysis which I did not attempt to perform in this paper.)
The key strategy is to study sets of the form
for various (which will usually be a non-zero-divisor), and some suitable threshold parameter M (which will be a little bit larger than the sum and product doubling constants of A). Roughly speaking, collects all the dilates of A which are “parallel” to the dilate in an additive sense. There is a remarkable “self-improving property” of these sets which shows, under suitable hypotheses on A, a, and M, that every element x of in fact obeys the improved estimate for some M’ that can be arranged to be significantly smaller than M. In other words, there is a gap phenomenon: the quantity can be very small or very large, but cannot be of intermediate size. This gap then leads to some very strong algebraic control on the ; for instance, under reasonable assumptions, becomes an additive group, and also exhibits good multiplicative closure properties (for instance, will be a subring). Once one has all this algebraic structure, the proofs of the theorems are then a relatively routine application of the sum-set theory and some elementary algebra.
There is still the issue of what to do when there are plenty of zero divisors. I don’t have a satisfactory resolution to this problem in general, but in the case of finite dimensional algebras over a field, in which the set of zero divisors forms an algebraic set, one can use some basic algebraic geometry to show that if a set of small additive doubling is concentrating inside the set of zero divisors, then it must in fact concentrate inside an affine subspace contained in the set of zero divisors. This does not fully determine the structure of such a set, but it seems to be a useful first step towards analysing this case further.
[See also my third Milliman lecture "Sum-product estimates, expanders, and exponential sums".]