About a year ago I wrote about the handsome theorem of Dummit and Hablicsek that there exist Besicovich sets over F_q[[t]]; that is, there are sets of measure 0 which contain a line in every direction. As I explained in that post, power series rings over finite fields are promising intermediate contexts for problems in additive combinatorics; like the real numbers, they have infinitely many scales, but unlike the real numbers, those scales naturally form a discrete set, and the whole ring decomposes in some iterative sense into a bunch of copies of a finite field, where things are much simpler (though by no means simple!)
But Dummit and Habliscek remained the only paper on additive combinatorics over non-archimedean local rings — until now! A new preprint by Bloom and Jones (Ph.D. students at Bristol) shows that if A is a finite subset of the ring of finite-headed Laurent series F_q((1/t)), then
The implicit constant depends on q, as it must, since one can get |A + A| = |A x A| = |A| by taking A to be the field of constants F_q itself.
This kind of sum-product problem has been the subject of sustained interest for a long time; the exponent is supposed to be 2 (i.e, either the sums are more or less distinct from each other or the products are) but nothing close to this has been obtained in any case. The exponent here is better than the best known for finite fields of prime order, but worse than the best known for the real numbers.
I haven’t read the paper yet, so I can’t say anything incisive about the methods, but it’s a striking result!