Partial difference equations over compact Abelian groups

Transcription

1 Partial difference equations over compact Abelian groups Tim Austin Courant Institute, NYU New York, NY, USA

2 SETTING Z a compact (metrizable) Abelian group U 1,..., U k closed subgroups of Z F(Z) {measurable functions Z T = R/Z} Z (up to equality a.e.) dz integral w.r.t. Haar probability measure If w Z, f F(Z) then d w f(z) := f(z w) f(z) discrete analog of directional derivative. If f : Z C, then w f(z) := f(z w)f(z) multiplicative variant

3 PARTIAL DIFFERENCE EQUATION Describe those f F(Z) for which d u1 d uk f(z) 0 u 1 U 1,..., u k U k. ZERO-SUM PROBLEM Describe those f 1,..., f k F(Z) s.t. and d ui f i (z) 0 u i U i, i = 1, 2,..., k f 1 (z) + f 2 (z) + + f k (z) 0. Will focus on PD ce Es in this talk.

4 MOTIVATION: SZEMERÉDI S THEOREM A k-ap in Z is a set of the form {z, z + r,..., z + (k 1)r} for some z, r Z. It is non-degenerate if all k entries are distinct. Theorem (Szemerédi 75). δ > 0 k 1 N 0 = N 0 (δ, k) 1 such that if N N 0 and E Z N with E = δn, then E some non-degenerate k-ap.

5 MULTI-DIMENSIONAL ANALOG Fix F = {v 1,..., v k } Z d, all v i distinct. If z Z d N and r Z N, let z + r F = {z + rv 1,..., z + rv k } mod N. Call this an F -constellation. It is non-degenerate if z + r F = k. Theorem (Furstenberg and Katznelson). δ > 0 d, k 1 N 0 = N 0 (δ, k, d) 1 such that if N N 0 and E Z d N with E = δn d then E some non-degenerate F -constn. Szemerédi s Theorem is special case F = {0, 1,..., k 1} Z.

6 Many proofs now known, using graph theory (Szemerédi), ergodic theory (Furstenberg), hypergraph theory (Nagle-Rödl-Schacht, Gowers, Tao) or harmonic analysis/additive combinatorics (Roth, Gowers). Roth s and Gowers harmonic-analysis proof gives much the best bound on N 0 (δ, k). Some proofs generalize to higher dimensions. Roth-Gowers approach has not been generalized, and the known dependence of N 0 (δ, k, d) is generally much worse when d 2.

7 SKETCH OF ROTH-GOWERS APPROACH Will formulate this for multi-dimensional theorem as far as possible. Important idea: estimate the fraction of all F -constellations that are contained in E. Let Z = Z d N. Suppose φ 1,..., φ k : Z C, and set Λ(φ 1,..., φ k ) := Z N Z d N φ 1 (z + rv 1 )φ 2 (z + rv 2 ) φ k (z + rv k ) dzdr. Interpretation: if E Z and φ 1 =... = φ k = 1 E, then Λ(1 E,..., 1 E ) = P ( random F -constn. lies in E ).

8 Idea is to show that if E = δn then Λ(1 E,..., 1 E ) const(δ, k) > 0. Since P ( randomly-chosen F -constn. is degenerate ) 0 as N, this proves the theorem.

9 Key tool for estimating Λ: directional Gowers uniformity norms. Can formulate these for any compact U 1,..., U k Z: If φ : Z C, then φ U(U1,...,U k ) := ( U 1 U 2 U k Z u1 uk φ(z) dzdu k du 1 ) 2 k.

10 For example, φ 4 U(U 1,U 2 ) = U 1 U 2 Z φ(z u 1 u 2 )φ(z u 1 )φ(z u 2 )φ(z) dzdu 1 du 2.

11 Theorem (Gowers-Cauchy-Schwartz inequality). Λ(φ 1,..., φ k ) φ 1 U(V12,...,V 1k ) φ 2 φ k, where V ij := Z N (v i v j ) Z d N for 1 i, j k. Proved by repeated use of Cauchy-Schwartz inequality. Corollary. If 1 E δ U(V12,...,V 1k ) 0, 1 E δ U(V21,V 23...,V 2k ) 0,. 1 E δ U(Vk1,...,V k,k 1 ) 0, then Λ(1 E,..., 1 E ) δ k, = many F -constellations in E if N is large.

12 Intuition: in this case, E behaves like a random set, and therefore contains many F -constellations by chance. Idea for full proof: if 1 E δ U(U1,...,U k 1 ) 0 for one of the lists of subgroups above, then deduce some special structure for E, which implies positive probability of F -constellations for a different reason (not by chance ). Proof is finished once have result for both random and structured E.

13 In one-dimension, have: Inverse Theorem for Gowers norms. In that case, V ij = Z N = Z for every i, j. Roughly: If φ 1 and φ U(Z,...,Z) η > 0, then a locally polynomial phase function ψ : Z S 1 s.t. Z φ(z)ψ(z) dz const(η) > 0. Won t define locally polynomial phase functions here.

14 Related toy problem: suppose φ(z) 1 z but Z Z u1 uk 1 φ(z) dzdu 1 du k 1 = 1. This is possible only if φ : Z S 1 and w1 wk 1 φ(z) 1. Exercise: if Z = Z N, N k and N prime, this occurs iff φ = exp(2πif/n) for some polynomial f : Z N Z N of degree k 2. There are more technical wrinkles for other Z, but still get some slightlygeneralized notion of polynomial of degree k 2.

15 In higher dimensions no good inverse theorem known for directional Gowers norms. Simplest toy case: describe those φ for which φ(z) 1 and φ U(U1,...,U k ) = 1. As before, this is equivalent to φ : Z S 1 and u1 uk φ 1 u 1 U 1,..., u k U k. Writing φ = exp(2πif) with f : Z T, this is exactly the partial difference equation we started with.

16 EXAMPLES OF PD ce Es AND SOLUTIONS Let us write our partial difference equation as d U 1 d U kf 0. Example 1 If U 1 =... = U k = Z, then this is the case of polynomial phase functions mentioned before. Example 2 There are always some obvious solutions: can take f = f f k, where d U if i = 0 (equivalently, f i is lifted from Z/U i T).

17 Definition. If f solves d U 1 d U kf 0, then let s say f is degenerate if can decompose it as f = f f k, where d U 2d U 3 d U kf 1 = d U 1d U 3 d U kf 2 =... = d U 1 d U k 1f k = 0. that is, each f i satisfies a simpler (and stronger) equation. One is interested in classifying non-degenerate solutions, modulo degenerate ones.

18 Example 3 On Z = T 3, let f(θ 1, θ 2, θ 3 ) = {θ 1 } + {θ 2 } θ 3. (For θ T, {θ} is its unique representative in [0, 1), and = integer part.) Then f(θ 1, θ 2, θ 3 ) f(θ 1, θ 2, θ 3 + θ 4 ) + f(θ 1, θ 2 + θ 3, θ 4 ) f(θ 1 + θ 2, θ 3, θ 4 ) + f(θ 2, θ 3, θ 4 ) = 0 Think of these as five different functions of (θ 1, θ 2, θ 3, θ 4 ) T 4. By differencing, we can annihilate the last four terms, leaving for suitable U 1,..., U 4 T 4. d U 1d U 2d U 3d U 4f = 0

19 f(θ 1, θ 2, θ 3 ) f(θ 1, θ 2, θ 3 + θ 4 ) + f(θ 1, θ 2 + θ 3, θ 4 ) f(θ 1 + θ 2, θ 3, θ 4 ) + f(θ 2, θ 3, θ 4 ) = 0 Disclosure: this equation is not arbitrary. It is the equation for a 3-cocycle in group cohomology H 3 m(t, T). One can prove that if f were degenerate, then it would be a coboundary: i.e., represent the zero class in H 3 m(t, T). But H 3 m(t, T) = Z, and I chose f above to be a generator. Hence nondegenerate.

20 More generally, H p m(t, T) = Z for all odd p. For each odd p, choosing a generator gives a non-degenerate solution to a PD ce E with p + 1 subgroups. This leads to many new cohomological examples.

21 Example 4 Let Z = T 2 T 2, and define σ, c : Z T by and σ(s, x) = {s 1 }{x 2 } {x 2 } + {s 2 } {x 1 + s 1 } mod 1 c(s, t) = {s 1 }{t 2 } {t 1 }{s 2 } mod 1 Then σ(s, x + t) σ(s, x) = σ(t, x + s) σ(t, x) + c(s, t). Interpret this as a zero-sum problem on T 2 T 2 T 2. As for Example 3, repeated differencing annihilates all but one term, leaving a new PD ce E for the subgroups U 1 = {(s, t, x) s = x + t = 0}, U 2 = {(s, t, x) s = x = 0}, U 3 = {(s, t, x) x + s = t = 0}, U 4 = {(s, t, x) s = t = 0}.

22 σ(s, x + t) σ(s, x) = σ(t, x + s) σ(t, x) + c(s, t). The reveal: Written multiplicatively (i.e., for maps Z S 1 ), this equation becomes σ(s, x + t) σ(s, x) σ(t, x + s) = c(s, t). σ(t, x) This is the Conze-Lesigne equation, important in the study of two-step nilrotations. Now a more complicated argument shows that σ is a non-degenerate solution to the PD ce E. Also: not obtained from a cocycle in group cohomology.

23 SOME POSITIVE RESULTS Assume U U k = Z. (Otherwise, just work on each coset of U U k separately.) Let M F(Z) be the subgroup of solutions to the PD ce E associated to U 1,..., U k. It is globally invariant under rotations of Z. Let M 0 M be the further subgroup of degenerate solutions. It is also globally Z-invariant. Lastly, let : T [0, 1/2] be distance from 0 in T.

24 Theorem (Small solutions are always degenerate). k 1 η > 0 such that f M and Z f < η = f M 0. Corollary. The quotient M/M 0 is a countable discrete group.

25 Theorem (Stability for approximate solutions). k 1 ε > 0 δ > 0 such that if f F(Z) and then g M U 1 s.t. U k Z d u 1 d uk f(z) < δ Z f g < ε.

26 Lastly, suppose Z = T d. A function f : T d T is step-polynomial if a partition such that Z = P 1 P m each P i is defined by linear inequalities in {θ i } for (θ 1,..., θ d ) T d, and each restriction f P i is some polynomial function of ({θ i }) d i=1, evaluated mod 1.

27 Theorem (Basic solns are step-polynomial, mod M 0 ). If f M, then f = f 1 + g for some step-polynomial f 1 M and some g M 0. Also, have bounds on complexity of f 1 that depend only on quantitative smoothness of f, not on choice of Z: this gives a nontrivial result even for Z finite.