The discrepancy of the linear flow on the torus Bence Borda Alfr´ ed R´ enyi Institute of Mathematics, Budapest Discrepancy Workshop, RICAM November 2018 Bence Borda The discrepancy of the linear flow on the torus
Discrepancy of curves The discrepancy of a sequence on the torus R d / Z d (= [0 , 1] d with opposite facets identified) cannot be O (1). (van Aardenne–Ehrenfest) is Ω(log N ) if d = 1 and Ω(log d / 2 N ) if d ≥ 2. (Schmidt, Roth) (From now on a N = Ω( b N ) means lim sup | a N | / b N > 0.) N →∞ Question: Is there a similar result for the discrepancy of curves? Given g : [0 , ∞ ) → R d / Z d and T > 0 let discrep ( g , T ) = sup | λ ( { 0 ≤ t ≤ T : g ( t ) ∈ R } ) − T λ ( R ) | R ∈R where R is the set of axis-parallel boxes in [0 , 1] d and λ is the Lebesgue measure. Under natural assumptions (e.g. if g is Lipschitz) we have discrep ( g , T ) = Ω(1). Is there a nontrivial lower estimate? Bence Borda The discrepancy of the linear flow on the torus
Curves with bounded discrepancy Drmota, 1989: Suppose g is continuous and has finite arc length ℓ T on any [0 , T ]. There exist curves for which discrep ( g , T ) = O (1) in d = 2. Conjectured that discrep ( g , T ) / T = Ω(log d − 2 − ε ℓ T /ℓ T ) in d ≥ 3. If g is Lipschitz, ℓ T = O ( T ) and so discrep ( g , T ) = Ω(log d − 2 − ε T ). We have recently proved that in fact there exist (Lipschitz) curves in any dimension d such that discrep ( g , T ) = O (1). As observed, this is best possible up to a constant factor. In particular, Drmota’s conjecture is false ; moreover, there is no van Aardenne–Ehrenfest type theorem for the discrep- ancy of curves in any dimension. Bence Borda The discrepancy of the linear flow on the torus
Linear flow Given α = ( α 1 , α 2 , . . . , α d ) ∈ R d the linear flow on the torus with direction α is the continuous time dynamical system which maps a point s ∈ R d / Z d to s + t α (mod Z d ) at time t ∈ R . For any function F : [0 , 1] d → R let � T � ∆ T ( s , α, F ) = F ( { s 1 + t α 1 } , . . . , { s d + t α d } ) d t − T [0 , 1] d F ( x ) d x . 0 For a set A ⊆ [0 , 1] d let ∆ T ( s , α, A ) = ∆ T ( s , α, χ A ) where χ A is the characteristic function of A . The following are equivalent: Every orbit is dense (Kronecker’s Theorem) Every orbit is uniformly distributed (Weyl’s Criterion) The dynamical system is ergodic α 1 , α 2 , . . . , α d are linearly independent over Q Bence Borda The discrepancy of the linear flow on the torus
Linear flow in dimension d = 2 Let α = ( α 1 , 1), 0 < α 1 < 1 irrational. Let � · � denote distance from the nearest integer. Assume � n α 1 � ≥ Cn − γ for every n ∈ N with some C > 0 and γ ≥ 1. Drmota, 1989: If γ < 2, then sup R ∈R | ∆ T ( s , α, R ) | = O (1). The proof used the Erd˝ os–Tur´ an inequality for curves. Grepstad–Larcher, 2016: If γ < 5 / 4, then ∆ T ( s , α, P ) = O (1) for any convex polygon P ⊆ [0 , 1] 2 whose sides are not parallel to α . The proof used an Ostrowski type explicit formula. B, 2018: If P ⊆ [0 , 1] 2 is a convex polygon with N sides, and φ 1 , φ 2 , . . . , φ N � = 0 , π are the angles of the sides and α , then ∞ | ∆ T ( s , α, P ) | ≤ 2 + N + 1 1 � 1 ≤ k <ℓ ≤ N | cot φ k − cot φ ℓ | max n 2 � n α 1 � . π 2 | α | n =1 Note γ < 2 holds for all algebraic irrationals. The exceptional set has Hausdorff dimension 2 / 3. Question: Is γ < 2 optimal? Bence Borda The discrepancy of the linear flow on the torus
Sketch of the proof Reduction to a discrete time dynamical system in dimension 1 (irrational rotation by α 1 ). Project the vertices along α to get 0 = c 0 < c 1 < · · · < c N +1 = 1. f ( x ) = a k x + b k on [ c k − 1 , c k ]. Slope a k depends on φ 1 , . . . , φ N . � 1 � c k N +1 N +1 e − 2 π inx f ( x ) e − 2 π inx d x = 1 � � ( telescoping ) − a k − 2 π in d x n 0 c k − 1 k =1 k =1 from integration by parts. The telescoping sum cancels by continuity of f , second sum is O (1 / n 2 ) with explicit implied constant depending only on N and φ 1 , . . . , φ N . Bence Borda The discrepancy of the linear flow on the torus
Linear flow in dimension d ≥ 3 Theorem (B, 2018) Let K be a subfield of R , α = ( α 1 , . . . , α d − 1 , 1) ∈ K d . Suppose that for any linearly independent linear forms L 1 , . . . , L d − 1 of d − 1 variables with coefficients in K there exist C > 0 and δ < 1 such that d − 1 � ( | L k ( n ) | + 1) ≥ C | n | − δ � n 1 α 1 + · · · + n d − 1 α d − 1 � k =1 for all n ∈ Z d − 1 , n � = 0 . Let P ⊆ [0 , 1] d be a polytope with nonempty interior such that every facet has a normal vector ν ∈ K d such that � ν, α � � = 0 . Then ∆ T ( s , α, P ) = O (1) with implied constant depending only on α and the normal vectors of the facets P. If K = algebraic reals, we get from Schmidt’s Subspace Theorem: Corollary (B, 2018) If the coordinates of α are algebraic and linearly independent over Q , then sup | ∆ T ( s , α, R ) | = O (1) . (Here R = set of axis parallel boxes.) R ∈R Bence Borda The discrepancy of the linear flow on the torus
Sketch of proof Reduction to a discrete time dynamical system in dimension d − 1 (irrational rotation by ( α 1 , α 2 , . . . , α d − 1 )). f : [0 , 1] d − 1 → R is still “piecewise linear”: project P along α to get a partition P 1 , . . . , P m of [0 , 1] d − 1 into polytopes. f ( x ) = � a k , x � + b k on P k . Here a k and the normal vectors of P k depend only on the normal vectors of P and α . � [0 , 1] d − 1 f ( x ) e − 2 π i � n , x � d x = m m � −� n , ν ( x ) � � a k , n � � f ( x ) e − 2 π i � n , x � d x + e − 2 π i � n , x � d x � � 2 π i | n | 2 2 π i | n | 2 ∂ P k P k k =1 k =1 from the Gauss–Ostrogradsky Theorem applied on each P k . The first sum cancels by the continuity of f (every facet shows up twice with opposite outer normal vectors ν ( x )). The Fourier coefficients of f are “smaller than expected” by a factor of | n | . Bence Borda The discrepancy of the linear flow on the torus
Questions We needed with some C > 0, δ < 1 d − 1 � ( | L k ( n ) | + 1) ≥ C | n | − δ . � n 1 α 1 + · · · + n d − 1 α d − 1 � k =1 Is δ < 1 optimal? If K = R , is the theorem true for almost every α ∈ R d ? The problem is that in the proof L 1 , . . . , L d − 1 not only depend on the normal vectors of P , but also on α . In d = 2: If γ < 5 / 4, the discrepancy with respect to balls is O (1). Convex sets with C 2 boundary of positive curvature are also sets of bounded remainder. (Grepstad–Larcher, 2016) If γ = 1 ( α 1 is badly approximable), the discrepancy with respect to all convex sets is O (log T ). Best possible up to a constant factor. f ( x ) is BV, reduces to Koksma’s inequality. (Beck, unpublished) Are there similar results in higher dimensions? Bence Borda The discrepancy of the linear flow on the torus
Discrete analogues For the discrete time dynamical system s �→ s + t α (mod Z d ), t ∈ Z the following are equivalent: Every orbit is dense Every orbit is uniformly distributed The dynamical system is ergodic 1 , α 1 , α 2 , . . . , α d are linearly independent over Q But the quantitative results are very different. Niederreiter, 1972: If 1 , α 1 , . . . , α d are algebraic and linearly independent over Q , then every orbit (=Kronecker sequence) has discrepancy O ( N ε ) for any ε > 0. Beck, 1994: Every orbit (=Kronecker sequence) has discrepancy O (log d N ϕ (log log N )) for a.e. α ∈ R d if and only if � ∞ n =1 1 /ϕ ( n ) < ∞ . (Here ϕ ( n ) > 0, increasing.) Bence Borda The discrepancy of the linear flow on the torus
Sets of bounded remainder asz, 1976: Let (Ω , F , µ, T ) be a discrete time, ergodic dynamical Hal´ system with µ (Ω) = 1. There exists a set of bounded remainder of measure 0 ≤ m ≤ 1 if and only if e 2 π im is an eigenvalue of the system (that is, there exists a measurable function g , not a.e. zero with g ( Tx ) = e 2 π im g ( x ) a.e.) If non-atomic, for any 2 ≤ ϕ (0) ≤ ϕ (1) ≤ · · · ≤ ϕ ( n ) → ∞ and any 0 ≤ m ≤ 1 there exists A ∈ F with µ ( A ) = m such that � n � � � � χ A ( T i x ) − n µ ( A ) � ≤ ϕ ( n ) a.e. � � � � � i =1 In particular, for the discrete time system s �→ s + t α (mod Z d ), t ∈ Z sets of bounded remainder have measure of the form n 0 + n 1 α 1 + · · · + n d α d where n 0 , n 1 , . . . , n d ∈ Z . In continuous time any measure 0 ≤ m ≤ 1 is possible (e.g. any axis-parallel box is a set of bounded remainder). Bence Borda The discrepancy of the linear flow on the torus
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