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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 Rd/Zd (= [0, 1]d with opposite facets identified) cannot be O(1). (van Aardenne–Ehrenfest) is Ω(log N) if d = 1 and Ω(logd/2 N) if d ≥ 2. (Schmidt, Roth) (From now on aN = Ω(bN) means lim sup

N→∞

|aN|/bN > 0.) Question: Is there a similar result for the discrepancy of curves? Given g : [0, ∞) → Rd/Zd and T > 0 let discrep(g, T) = sup

R∈R

|λ ({0 ≤ t ≤ T : g(t) ∈ R}) − Tλ(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 ℓT on any [0, T]. There exist curves for which discrep(g, T) = O(1) in d = 2. Conjectured that discrep(g, T)/T = Ω(logd−2−ε ℓT/ℓT) in d ≥ 3. If g is Lipschitz, ℓT = O(T) and so discrep(g, T) = Ω(logd−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 = (α1, α2, . . . , αd) ∈ Rd the linear flow on the torus with direction α is the continuous time dynamical system which maps a point s ∈ Rd/Zd to s + tα (mod Zd) at time t ∈ R. For any function F : [0, 1]d → R let ∆T(s, α, F) = T F({s1 + tα1}, . . . , {sd + tαd}) dt − T

• [0,1]d F(x) dx.

For a set A ⊆ [0, 1]d let ∆T(s, α, A) = ∆T(s, α, χA) where χA 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 α1, α2, . . . , αd are linearly independent over Q

Bence Borda The discrepancy of the linear flow on the torus

Linear flow in dimension d = 2

Let α = (α1, 1), 0 < α1 < 1 irrational. Let · denote distance from the nearest integer. Assume nα1 ≥ Cn−γ for every n ∈ N with some C > 0 and γ ≥ 1. Drmota, 1989: If γ < 2, then supR∈R |∆T(s, α, R)| = O(1). The proof used the Erd˝

• s–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 φ1, φ2, . . . , φN = 0, π are the angles of the sides and α, then |∆T(s, α, P)| ≤ 2 + N + 1 π2|α| max

1≤k<ℓ≤N | cot φk − cot φℓ| ∞

• n=1

1 n2nα1. 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 α1). Project the vertices along α to get 0 = c0 < c1 < · · · < cN+1 = 1. f (x) = akx + bk on [ck−1, ck]. Slope ak depends on φ1, . . . , φN. 1 f (x)e−2πinx dx = 1 n

N+1

• k=1

(telescoping) −

N+1

• k=1

ak ck

ck−1

e−2πinx −2πin dx from integration by parts. The telescoping sum cancels by continuity of f , second sum is O(1/n2) with explicit implied constant depending only

• n N and φ1, . . . , φN.

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 = (α1, . . . , αd−1, 1) ∈ K d. Suppose that for any linearly independent linear forms L1, . . . , Ld−1 of d − 1 variables with coefficients in K there exist C > 0 and δ < 1 such that n1α1 + · · · + nd−1αd−1

d−1

• k=1

(|Lk(n)| + 1) ≥ C|n|−δ for all n ∈ Zd−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

• nly 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

R∈R

|∆T(s, α, R)| = O(1). (Here R = set of axis parallel boxes.)

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 (α1, α2, . . . , αd−1)). f : [0, 1]d−1 → R is still “piecewise linear”: project P along α to get a partition P1, . . . , Pm of [0, 1]d−1 into polytopes. f (x) = ak, x + bk on Pk. Here ak and the normal vectors of Pk depend only on the normal vectors of P and α.

• [0,1]d−1 f (x)e−2πin,x dx =

m

• k=1
• ∂Pk

−n, ν(x) 2πi|n|2 f (x)e−2πin,x dx+

m

• k=1

ak, n 2πi|n|2

• Pk

e−2πin,x dx from the Gauss–Ostrogradsky Theorem applied on each Pk. 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

• f |n|.

Bence Borda The discrepancy of the linear flow on the torus

Questions

We needed with some C > 0, δ < 1 n1α1 + · · · + nd−1αd−1

d−1

• k=1

(|Lk(n)| + 1) ≥ C|n|−δ. Is δ < 1 optimal? If K = R, is the theorem true for almost every α ∈ Rd? The problem is that in the proof L1, . . . , Ld−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 (α1 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 + tα (mod Zd), t ∈ Z the following are equivalent: Every orbit is dense Every orbit is uniformly distributed The dynamical system is ergodic 1, α1, α2, . . . , αd are linearly independent over Q But the quantitative results are very different. Niederreiter, 1972: If 1, α1, . . . , αd are algebraic and linearly independent

• ver Q, then every orbit (=Kronecker sequence) has discrepancy O(Nε)

for any ε > 0. Beck, 1994: Every orbit (=Kronecker sequence) has discrepancy O(logd Nϕ(log log N)) for a.e. α ∈ Rd 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

Hal´ asz, 1976: Let (Ω, F, µ, T) be a discrete time, ergodic dynamical system with µ(Ω) = 1. There exists a set of bounded remainder of measure 0 ≤ m ≤ 1 if and only if e2πim is an eigenvalue of the system (that is, there exists a measurable function g, not a.e. zero with g(Tx) = e2πimg(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
• i=1

χA(T ix) − nµ(A)

• ≤ ϕ(n)

a.e. In particular, for the discrete time system s → s + tα (mod Zd), t ∈ Z sets of bounded remainder have measure of the form n0+n1α1+· · ·+ndαd where n0, n1, . . . , nd ∈ 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

Measurable test functions

Both discrete and continuous time: given α (with the appropriate linear independence property) and F ∈ L1([0, 1]d), we have ∆T(s, α, F) = o(T) for a.e. s ∈ [0, 1]d. (Birkhoff) What if s ∈ [0, 1]d and F are given, and we want metric results in α? In discrete time there exists an open set A ⊆ [0, 1] with ∆T(0, α, A) = Ω(T) for every α. (Marstrand’s counterexample to Khinchin’s Conjecture) In continuous time for any F ∈ L2([0, 1]d) we have ∆T(0, α, F) = O(T 1/2−1/(2d−2) log3+ε T) for a.e. α ∈ Rd. Optimal up to powers of log T. (Beck, 2015) Question: Estimate optimal up to a constant factor? This would be most interesting in d = 2 when ∆T(0, α, F) = O(log3+ε T) for a.e. α ∈ R2.

Bence Borda The discrepancy of the linear flow on the torus

Thank you!

References

[1] J. Beck: From Khinchin’s conjecture on strong uniformity to superuniform motions. Mathematika 61 (2015), no. 3, 591–707 [2] B. Borda: Bounded error uniformity of the linear flow on the torus. to appear in

• Monatsh. Math. (also available on arxiv)

[3] M. Drmota: Irregularities of continuous distributions. Ann. Inst. Fourier (Grenoble) 39 (1989), no. 3, 501–527 [4] S. Grepstad, G. Larcher: Sets of bounded remainder for a continuous irrational rotation on [0, 1]2. Acta Arith. 176 (2016), no. 4, 365–395 [5] G. Hal´ asz: Remarks on the remainder in Birkhoff’s ergodic theorem. Acta Math.

• Acad. Sci. Hungar. 28 (1976) 389–395

[6] H. Niederreiter: Methods for estimating discrepancy. Applications of number theory to numerical analysis (Proc. Sympos., Univ. Montr´ eal, Montreal Que., 1971), pp. 203–236. Academic Press, New York, 1972.

Bence Borda The discrepancy of the linear flow on the torus