An Ultrafilter Approach to Jin’s Theorem

It is well known and not difficult to prove that if C ⊆ Z has positive upper Banach density, the set of differences C −C is syndetic, i.e. the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed that whenever A and B have positive upper Banach density, then A − B is piecewise syndetic. Jin’s result follows trivially from the first statement provided that B has large intersection with a shifted copy A−n of A. Of course this will not happen in general if we consider shifts by integers, but the idea can be put to work if we allow “shifts by ultrafilters”. As a consequence we obtain Jin’s Theorem. The upper Banach density of C ⊆ Z is given by d(C) := limm−n→∞ 1 m−n+1 |C∩{n, . . . ,m}|. A set S ⊆ Z is syndetic if the gaps of S are of uniformly bounded length, i.e. if there exists some k > 0 such that S −{−k, . . . , k} = Z. A set P ⊆ Z is piecewise syndetic if it is syndetic on large pieces, i.e. if there exists some k ≥ 0 such that P − {−k, . . . , k} contains arbitrarily long intervals of integers. It was first noted by Følner ([Føl54a, Føl54b]) that C − C = {c1 − c2 : c1, c2 ∈ C} is syndetic provided that d(C) > 0. (To see this, pick a subset {i1, . . . , im} ⊆ Z which is maximal subject to the condition that C − i1, . . . ,C − im are mutually disjoint. This is possible since disjointness implies d(C−i1∪. . .∪C−im) = m·d(C), therefore m is at most 1/d(C). But then maximality implies that for each n ∈ Z there is some ik, k ∈ {1, . . . ,m} such that (C − n) ∩ (C − ik) , ∅, resp. n ∈ (C − C) + ik. Thus ⋃m k=1(C − C) + ik = Z.) Simple counterexamples yield that the analogous statement fails when two different sets are considered, but Renling Jin discovered the following interesting result. Theorem 1 ([Jin02]). Let A, B ⊆ Z, d(A), d(B) > 0. Then A − B is piecewise syndetic. In Section 1 we reprove Jin’s Theorem by reducing it to the C − C case. Subsequently we discuss some modifications of our argument which allow to recover the refinements resp. generalizations of Jin’s result found in [JK03, BFW06, Jin08, BBF]. 1 Ultrafilter proof of Jin’s Theorem As indicated in the abstract, we aim to show that one can shift a given large set A ⊆ Z by an ultrafilter so that it will have large intersection with another, previously specified set. We motivate the definition of this ultrafilter-shift by means of analogy: For n ∈ Z denote by e(n) the principle ultrafilter on Z which corresponds to n and notice that A − n = {k ∈ Z : n ∈ A − k} = {k ∈ Z : A − k ∈ e(n)}. Given an ultrafilter p on Z, we thus define {k ∈ Z : A − k ∈ p} as the official meaning of “A − p”. 2000 Mathematics Subject Classification. 11B05, 05D10.