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 U 1 , . . . , U k closed subgroups of Z F ( Z ) { measurable functions Z − → T = R / Z } (up to equality a.e.) � · d z integral w.r.t. Haar probability measure Z 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

PARTIAL DIFFERENCE EQUATION Describe those f ∈ F ( Z ) for which d u 1 · · · d u k f ( z ) ≡ 0 ∀ u 1 ∈ U 1 , . . . , u k ∈ U k . ZERO-SUM PROBLEM Describe those f 1 , . . . , f k ∈ F ( Z ) s.t. d u i f i ( z ) ≡ 0 ∀ u i ∈ U i , i = 1 , 2 , . . . , k and f 1 ( z ) + f 2 ( z ) + · · · + f k ( z ) ≡ 0 . Will focus on PD ce Es 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 ∃ 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 .

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 N with | E | = δN d then E ⊆ Z d 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¨ odl-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 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 = Z d N . Suppose φ 1 , . . . , φ k : Z − → C , and set Λ( φ 1 , . . . , φ k ) := � � φ 1 ( z + rv 1 ) φ 2 ( z + rv 2 ) · · · φ k ( z + rv k ) d z d r. Z d Z N N Interpretation: if E ⊆ Z and φ 1 = . . . = φ k = 1 E , then � � Λ(1 E , . . . , 1 E ) = P random F -constn. lies in E .

Idea is to show that if | E | = δN then Λ(1 E , . . . , 1 E ) ≥ const( δ, k ) > 0 . Since � � → 0 P randomly-chosen F -constn. is degenerate − as N − → ∞ , this proves the theorem.

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 � 2 − k � � � � � � φ � U( U 1 ,...,U k ) := · · · ∇ u 1 · · · ∇ u k φ ( z ) d z d u k · · · d u 1 . U 1 U 2 U k Z

For example, � φ � 4 U( U 1 ,U 2 ) = � � � φ ( z − u 1 − u 2 ) φ ( z − u 1 ) φ ( z − u 2 ) φ ( z ) d z d u 1 d u 2 . U 1 U 2 Z

Theorem (Gowers-Cauchy-Schwartz inequality) . | Λ( φ 1 , . . . , φ k ) | ≤ � φ 1 � U( V 12 ,...,V 1 k ) · � 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( V 12 ,...,V 1 k ) ≈ 0 , � 1 E − δ � U( V 21 ,V 23 ...,V 2 k ) ≈ 0 , . . . � 1 E − δ � U( V k 1 ,...,V k,k − 1 ) ≈ 0 , then Λ(1 E , . . . , 1 E ) ≈ δ k , = ⇒ 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 � 1 E − δ � U( U 1 ,...,U k − 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, V ij = Z N = Z for every i, j . Roughly: If � φ � ∞ ≤ 1 and � φ � U( Z,...,Z ) ≥ η > 0 , then ∃ a ‘locally polyno- → S 1 s.t. mial phase function’ ψ : Z − � � � � � φ ( z ) ψ ( z ) d z � ≥ const( η ) > 0 . � � � Z Won’t define ‘locally polynomial phase functions’ here.

Related toy problem: suppose | φ ( z ) | ≤ 1 ∀ z but � � · · · ∇ u 1 · · · ∇ u k − 1 φ ( z ) d z d u 1 · · · d u k − 1 = 1 . Z Z → S 1 and This is possible only if φ : Z − ∇ w 1 · · · ∇ w k − 1 φ ( z ) ≡ 1 . Exercise: if Z = Z N , N ≫ k and N prime, this occurs iff φ = exp(2 π i f/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 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 � φ � U( U 1 ,...,U k ) = 1 . → S 1 and As before, this is equivalent to φ : Z − ∇ u 1 · · · ∇ u k φ ≡ 1 ∀ u 1 ∈ U 1 , . . . , u k ∈ U k . Writing φ = exp(2 π i f ) with f : Z − → T , this is exactly the partial differ- ence equation we started with.

EXAMPLES OF PD ce Es AND SOLUTIONS Let us write our partial difference equation as d U 1 · · · d U k f ≡ 0 . If U 1 = . . . = U k = Z , then this is the case of ‘polynomial Example 1 phase functions’ mentioned before. Example 2 There are always some obvious solutions: can take f = f 1 + . . . + f k , where d U i f i = 0 (equivalently, f i is lifted from Z/U i − → T ).

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

On Z = T 3 , let f ( θ 1 , θ 2 , θ 3 ) = ⌊{ θ 1 } + { θ 2 }⌋ · θ 3 . Example 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 d U 1 d U 2 d U 3 d U 4 f = 0 for suitable U 1 , . . . , U 4 ≤ T 4 .

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 ) . m ( T , T ) ∼ But H 3 = Z , and I chose f above to be a generator. Hence non- degenerate.

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 sub- groups. This leads to many new ‘cohomological’ examples.

Let Z = T 2 × T 2 , and define σ, c : Z − Example 4 → T by σ ( s, x ) = { s 1 }{ x 2 } − ⌊{ x 2 } + { s 2 }⌋{ x 1 + s 1 } mod 1 and 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, leav- ing 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 } .

σ ( s, x + t ) − σ ( s, x ) = σ ( t, x + s ) − σ ( t, x ) + c ( s, t ) . The reveal: → S 1 ), this equation becomes Written multiplicatively (i.e., for maps Z − σ ( s, x + t ) = c ( s, t ) σ ( t, x + s ) . σ ( s, x ) 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 PD ce E. Also: not obtained from a cocycle in group cohomology.

SOME POSITIVE RESULTS Assume U 1 + . . . + U k = Z . (Otherwise, just work on each coset of U 1 + . . . + 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 .