Convergence Results for Systems of Linear Forms on Cyclic Groups and Periodic Nilsequences

Abstract

Given a positive integer N and real number α ∈ [0, 1], let m(α, N ) denote the minimum, over all sets A ⊆ Z N of size at least αN , of the normalized count of 3-term arithmetic progressions contained in A. A theorem of Croot states that m(α, N ) converges as N → ∞ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for k-term progressions for general k and further for all systems of integer linear forms of finite complexity. We also obtain a similar convergence result for the maximum densities of sets free of solutions to systems of linear equations. These results rely on a regularity method for functions on finite cyclic groups that we frame in terms of periodic nilsequences, using in particular some regularity results of Szegedy (relying on his joint work with Camarena) and the equidistribution results of Green and Tao.

A continuous model for systems of complexity 2 on simple abelian groups

“…This is a strong form of equidistribution in the sense that, in addition to the given map being equidistributed, we require various multiparameter versions of the map, corresponding to Leibman nilmanifolds for systems of linear forms, also to be equidistributed. In the special case of polynomial sequences, this is related to the stronger notion of irrationality from [13] (see also [2]).…”

“…, ψ t ) be a system of integer linear forms, thus ψ i : Z D → Z is a homomorphism for each i ∈ [t]. For a compact abelian group Z and a measurable 2 We recall this notion of complexity in Definition A.7. Let us recall also that the infimum in m k (α, Z N )…”

“…can be restricted to indicator functions of subsets of Z N without affecting its asymptotic behaviour, as explained in [3, p. 3] for k = 3; for k > 3 one can argue similarly, using [2,Lemma 8.2].…”

“…Theorem 1.5. Let X 2 = (T 2 , R •(1) × R • (2) ). Then for every system Ψ of integer linear forms of complexity at most 2, and every α ∈ [0, 1], we have lim p→∞ m Ψ (α, Z p ) = m Ψ (α, X 2 ).…”

“…On X 1 , the Leibman nilmanifold corresponding to Ψ 4 is the 'usual' 2-dimensional subgroup of T 4 consisting of all 4-term progressions: X Ψ 4 1 = {(x 1 , x 1 +x 2 , x 1 +2x 2 , x 1 +3x 2 ) : x 1 , x 2 ∈ T}. If instead we equip T with the filtration R • (2) , letting Y = (T, R •(2) ), then the corresponding Leibman nilmanifold for Ψ 4 is the following 3-dimensional subgroup of T 4 , which contains X Ψ 4 1 : Y Ψ 4 = {(y 1 , y 1 + y 2 , y 1 + 2y 2 + y 3 , y 1 + 3y 2 + 3y 3 ) : y i ∈ T}. (Similar calculations have been used as examples in previous works; see in particular [13,Section 3].)…”