Partial difference equations over compact Abelian groups Tim Austin - - PowerPoint PPT Presentation

Partial difference equations over compact Abelian groups

Tim Austin Courant Institute, NYU New York, NY, USA tim@cims.nyu.edu

SETTING Z a compact (metrizable) Abelian group U1, . . . , Uk closed subgroups of Z F(Z) {measurable functions Z − → T = R/Z}

(up to equality a.e.)

• Z

· dz integral w.r.t. Haar probability measure If w ∈ Z, f ∈ F(Z) then dwf(z) := f(z − w) − f(z) — discrete analog of directional derivative. If f : Z − → C, then ∇wf(z) := f(z − w)f(z) — multiplicative variant

PARTIAL DIFFERENCE EQUATION Describe those f ∈ F(Z) for which du1 · · · dukf(z) ≡ 0 ∀u1 ∈ U1, . . . , uk ∈ Uk. ZERO-SUM PROBLEM Describe those f1, . . . , fk ∈ F(Z) s.t. duifi(z) ≡ 0 ∀ui ∈ Ui, i = 1, 2, . . . , k and f1(z) + f2(z) + · · · + fk(z) ≡ 0. Will focus on PDceEs in this talk.

MOTIVATION: SZEMER ´ EDI’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 (Szemer´ edi ’75). ∀δ > 0 ∀k ≥ 1 ∃N0 = N0(δ, k) ≥ 1 such that if N ≥ N0 and E ⊆ ZN with |E| = δN, then E ⊇ some non-degenerate k-AP.

MULTI-DIMENSIONAL ANALOG Fix F = {v1, . . . , vk} ⊂ Zd, all vi distinct. If z ∈ Zd

N and r ∈ ZN, let

z + r · F = {z + rv1, . . . , z + rvk} 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 ∃N0 = N0(δ, k, d) ≥ 1 such that if N ≥ N0 and E ⊆ Zd

N with |E| = δNd then

E ⊇ some non-degenerate F-constn. Szemer´ edi’s Theorem is special case F = {0, 1, . . . , k − 1} ⊂ Z.

Many proofs now known, using graph theory (Szemer´ edi), ergodic theory (Furstenberg), hypergraph theory (Nagle-R¨

• dl-Schacht, Gowers, Tao) or

harmonic analysis/additive combinatorics (Roth, Gowers). Roth’s and Gowers’ harmonic-analysis proof gives much the best bound

• n N0(δ, k).

Some proofs generalize to higher dimensions. Roth-Gowers approach has not been generalized, and the known dependence of N0(δ, k, d) is gener- ally much worse when d ≥ 2.

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 con- tained in E. Let Z = Zd

N.

Suppose φ1, . . . , φk : Z − → C, and set Λ(φ1, . . . , φk) :=

• ZN
• Zd

N

φ1(z + rv1)φ2(z + rv2) · · · φk(z + rvk) dzdr. Interpretation: if E ⊆ Z and φ1 = . . . = φk = 1E, then Λ(1E, . . . , 1E) = P

• random F-constn. lies in E
• .

Idea is to show that if |E| = δN then Λ(1E, . . . , 1E) ≥ const(δ, k) > 0. Since P

• randomly-chosen F-constn. is degenerate
• −

→ 0 as N − → ∞, this proves the theorem.

Key tool for estimating Λ: directional Gowers uniformity norms. Can formulate these for any compact U1, . . . , Uk ≤ Z: If φ : Z − → C, then φU(U1,...,Uk) :=

U1

• U2

· · ·

• Uk
• Z

∇u1 · · · ∇ukφ(z) dzduk · · · du1

2−k

.

For example, φ4

U(U1,U2) =

• U1
• U2
• Z

φ(z − u1 − u2)φ(z − u1)φ(z − u2)φ(z) dzdu1du2.

Theorem (Gowers-Cauchy-Schwartz inequality). |Λ(φ1, . . . , φk)| ≤ φ1U(V12,...,V1k) · φ2∞ · · · φk∞, where Vij := ZN · (vi − vj) ≤ Zd

N for 1 ≤ i, j ≤ k.

Proved by repeated use of Cauchy-Schwartz inequality.

• Corollary. If

1E − δU(V12,...,V1k) ≈ 0, 1E − δU(V21,V23...,V2k) ≈ 0, . . . 1E − δU(Vk1,...,Vk,k−1) ≈ 0, then Λ(1E, . . . , 1E) ≈ δk, = ⇒ many F-constellations in E if N is large.

Intuition: in this case, E ‘behaves like a random set’, and therefore ‘con- tains many F-constellations by chance’. Idea for full proof: if 1E − δU(U1,...,Uk−1) ≈ 0 for one of the lists of subgroups above, then deduce some special ‘struc- ture’ for E, which implies positive probability of F-constellations for a dif- ferent reason (not ‘by chance’). Proof is finished once have result for both ‘random’ and ‘structured’ E.

In one-dimension, have: Inverse Theorem for Gowers norms. In that case, Vij = ZN = Z for every i, j. Roughly: If φ∞ ≤ 1 and φU(Z,...,Z) ≥ η > 0, then ∃ a ‘locally polyno- mial phase function’ ψ : Z − → S1 s.t.

• Z

φ(z)ψ(z) dz

• ≥ const(η) > 0.

Won’t define ‘locally polynomial phase functions’ here.

Related toy problem: suppose |φ(z)| ≤ 1 ∀z but

• Z

· · ·

• Z

∇u1 · · · ∇uk−1φ(z) dzdu1 · · · duk−1 = 1. This is possible only if φ : Z − → S1 and ∇w1 · · · ∇wk−1φ(z) ≡ 1. Exercise: if Z = ZN, N ≫ k and N prime, this occurs iff φ = exp(2πif/N) for some polynomial f : ZN − → ZN of degree ≤ k − 2. There are more technical wrinkles for other Z, but still get some slightly- generalized notion of ‘polynomial of degree ≤ k − 2’.

In higher dimensions no good inverse theorem known for directional Gow- ers norms. Simplest toy case: describe those φ for which |φ(z)| ≤ 1 and φU(U1,...,Uk) = 1. As before, this is equivalent to φ : Z − → S1 and ∇u1 · · · ∇ukφ ≡ 1 ∀u1 ∈ U1, . . . , uk ∈ Uk. Writing φ = exp(2πif) with f : Z − → T, this is exactly the partial differ- ence equation we started with.

EXAMPLES OF PDceEs AND SOLUTIONS Let us write our partial difference equation as dU1 · · · dUkf ≡ 0. Example 1 If U1 = . . . = Uk = Z, then this is the case of ‘polynomial phase functions’ mentioned before. Example 2 There are always some obvious solutions: can take f = f1 + . . . + fk, where dUifi = 0 (equivalently, fi is lifted from Z/Ui − → T).

• Definition. If f solves

dU1 · · · dUkf ≡ 0, then let’s say f is degenerate if can decompose it as f = f1 + · · · + fk, where dU2dU3 · · · dUkf1 = dU1dU3 · · · dUkf2 = . . . = dU1 · · · dUk−1fk = 0. – that is, each fi satisfies a simpler (and stronger) equation. One is interested in classifying non-degenerate solutions, modulo degen- erate ones.

Example 3 On Z = T3, let f(θ1, θ2, θ3) = ⌊{θ1} + {θ2}⌋ · θ3.

(For θ ∈ T, {θ} is its unique representative in [0, 1), and ⌊·⌋ = integer part.)

Then f(θ1, θ2, θ3) − f(θ1, θ2, θ3 + θ4) + f(θ1, θ2 + θ3, θ4) − f(θ1 + θ2, θ3, θ4) + f(θ2, θ3, θ4) = 0 Think of these as five different functions of (θ1, θ2, θ3, θ4) ∈ T4. By differencing, we can annihilate the last four terms, leaving dU1dU2dU3dU4f = 0 for suitable U1, . . . , U4 ≤ T4.

f(θ1, θ2, θ3) − f(θ1, θ2, θ3 + θ4) + f(θ1, θ2 + θ3, θ4) − f(θ1 + θ2, θ3, θ4) + f(θ2, θ3, θ4) = 0 Disclosure: this equation is not arbitrary. It is the equation for a 3-cocycle in group cohomology H3

m(T, T).

One can prove that if f were degenerate, then it would be a coboundary: i.e., represent the zero class in H3

m(T, T).

But H3

m(T, T) ∼

= Z, and I chose f above to be a generator. Hence non- degenerate.

More generally, Hp

m(T, T) ∼

= Z for all odd p. For each odd p, choosing a generator gives a non-degenerate solution to a PDceE with p + 1 sub- groups. This leads to many new ‘cohomological’ examples.

Example 4 Let Z = T2 × T2, and define σ, c : Z − → T by σ(s, x) = {s1}{x2} − ⌊{x2} + {s2}⌋{x1 + s1} mod 1 and c(s, t) = {s1}{t2} − {t1}{s2} 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 T2 × T2 × T2. As for Example 3, repeated differencing annihilates all but one term, leav- ing a new PDceE for the subgroups U1 = {(s, t, x) | s = x + t = 0}, U2 = {(s, t, x) | s = x = 0}, U3 = {(s, t, x) | x + s = t = 0}, U4 = {(s, t, x) | s = t = 0}.

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 − → S1), this equation becomes σ(s, x + t) σ(s, x) = c(s, t)σ(t, x + s) 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 solu- tion to the PDceE. Also: not obtained from a cocycle in group cohomology.

SOME POSITIVE RESULTS Assume U1 + . . . + Uk = Z. (Otherwise, just work on each coset of U1 + . . . + Uk separately.) Let M ⊆ F(Z) be the subgroup of solutions to the PDceE associated to U1, . . . , Uk. It is globally invariant under rotations of Z. Let M0 ⊆ 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.

Theorem (Small solutions are always degenerate). ∀k ≥ 1 ∃η > 0 such that f ∈ M and

• Z |f| < η

= ⇒ f ∈ M0.

• Corollary. The quotient M/M0 is a countable discrete group.

Theorem (Stability for approximate solutions). ∀k ≥ 1 ∀ε > 0 ∃δ > 0 such that if f ∈ F(Z) and

• U1

· · ·

• Uk
• Z |du1 · · · dukf(z)| < δ

then ∃g ∈ M s.t.

• Z |f − g| < ε.

Lastly, suppose Z = Td. A function f : Td − → T is step-polynomial if ∃ a partition Z = P1 ∪ · · · ∪ Pm such that

• each Pi is defined by linear inequalities in {θi} for (θ1, . . . , θd) ∈ Td,

and

• each restriction f|Pi is some polynomial function of ({θi})d

i=1, evalu-

ated mod 1.

Strong expectation: Solns are step-polynomial, mod M0. If f ∈ M, then f = f1 + g for some step-polynomial f1 ∈ M and some g ∈ M0. Also, have bounds

• n ‘complexity’ of f1 that depend only on ‘quantitative smoothness’ of f,

not on choice of Z: this gives a nontrivial result even for Z finite. (This result is work in progress.)