In this paper we look for conditions that are sufficient to guarantee that a subset A of a finite Abelian group G contains the ‘expected’ number of linear configurations of a given type. The simplest non‐trivial result of this kind is the well‐known fact that if G has odd order, A has density α and all Fourier coefficients of the characteristic function of A are significantly smaller than α (except the one at zero, which equals α), then A contains approximately α3|G|2 triples of the form (a, a+d, a+2d). This is ‘expected’ in the sense that a random set A of density α has approximately α3|G|2 such triples with very high probability. More generally, it was shown by the first author (in the case G = ℤN for N prime, but the proof generalizes) that a set A of density α has about αk|G|2 arithmetic progressions of length k if the characteristic function of A is almost as small as it can be, given its density, in a norm that is now called the Uk−1‐norm. When investigating linear equations in the primes, Green and Tao found the most general statement that follows from the technique used to prove this result, introducing a notion that they call the complexity of a system of linear forms. They prove that if A has almost minimal Uk+1‐norm, then it has the expected number of linear configurations of a given type, provided that the associated complexity is at most k. The main result of this paper is that the converse is not true: in particular there are certain systems of complexity 2 that are controlled by the U2‐norm, whereas the result of Green and Tao requires the stronger hypothesis of U3‐control. We say that a system of m linear forms L1, …, Lm in d variables with integer coeffcients has true complexity k if k is the smallest positive integer such that, for any set A of density α and almost minimal Uk+1‐norm, the number of d‐tuples (x1, …, xd) such that Li(x1, …, xd) ∈ A for every i is approximately αm|G|d. We conjecture that the true complexity k is the smallest positive integer s for which the functions L1s+1, … ,Lms+1 are linearly independent. Using the ‘quadratic Fourier analysis’ of Green and Tao we prove this conjecture in the case where the complexity of the system (in Green and Tao's sense) is 2, s=1 and G is the group 𝔽pn for some fixed odd prime p. A closely related result in ergodic theory was recently proved independently by Leibman. We end the paper by discussing the connections between his result and ours.

Although uncommon, nonvalvular infections of the cardiovascular system will increase in frequency as the use of implantable devices and prosthetic materials increases in the elderly. Studies are needed to determine the most appropriate diagnostic methods, treatment regimens, and methods for prevention of these infections.

In [GW1] we began an investigation of the following general question. Let L 1 , . . . , L m be a system of linear forms in d variables on F n p , and let A be a subset of F n p of positive density. Under what circumstances can one prove that A contains roughly the same number of m-tuples L 1 (x 1 , . . . , x d ), . . . , L m (x 1 , . . . , x d ) with x 1 , . . . , x d ∈ F n p as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that A − δ1 U k should be small, where we have written A for the characteristic function of the set A, δ is the density of A, k is some parameter that depends on the linear forms L 1 , . . . , L m , and . U k is the kth uniformity norm. The question we investigated was how k depends on L 1 , . . . , L m . Our main result was that there were systems of forms where k could be taken to be 2 even though there was no simple proof of this fact using the Cauchy-Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree k − 1 is a sufficient condition if and only if the kth powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that p is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the U k norm over F n p by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.

The reductive dehalogenation of chlorinated propenes was studied with the tetrachloroethene reductive dehalogenase purified from Sulfurospirillum multivorans to obtain indications for a radical mechanism of this reaction. When reduced methyl viologen (MV), which is a radical cation, was applied as electron donor for the reduction of different chloropropenes, a significant part of MV could not be rereduced with Ti(III) citrate, indicating that a part of the MV was consumed in a side reaction. Mass spectrometric analysis of assays with MV as electron donor revealed the formation of side products, the masses of which might account for the formation of adducts from a chloropropenyl radical and reduced methyl viologen. With Ti(III) citrate as sole electron donor, 2,3-dichloropropene was reduced and as a side product, 2,5-dichloro-1,5-hexadiene was formed demonstrating that the reductive dechlorination of 2,3-dichloropropene proceeds via a radical reaction mechanism. The results support different dehalogenation mechanisms forthe reductive dechlorination of chloropropenes and halogenated ethenes.

The arithmetic regularity lemma for Fpn, proved by Green in 2005, states that given a subset A⊆double-struckFpn, there exists a subspace H⩽double-struckFpn of bounded codimension such that A is Fourier‐uniform with respect to almost all cosets of H. It is known that in general, the growth of the codimension of H is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non‐uniform cosets. Our main result is that, under a natural model‐theoretic assumption of stability, the tower‐type bound and non‐uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we say that a set A⊆double-struckFpn is k‐stable if there are no a1,…,ak,b1,…,bk∈double-struckFpn such that ai+bj∈A if and only if i⩽j. We prove an arithmetic regularity lemma for k‐stable subsets A⊆double-struckFpn in which the bound on the codimension of the subspace is a polynomial (depending on k) in the degree of uniformity, and in which there are no non‐uniform cosets. This result is an arithmetic analogue of the stable graph regularity lemma proved by Malliaris and Shelah.

Abstract. We provide a short proof of a recent result of Elkin in which large subsets of {1, . . . , N } free of 3-term progressions are constructed.

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form G = F n p , provided p is not too small. In Z N , substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm . U 3 requires a more sophisticated (local) formulation. When N is prime, Z N is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].

No abstract