On the convergence to 0 of m_n x [1].

ON THE CONVERGENCE TO0 OF mnξ mod 1

BASSAM FAYAD AND JEAN-PAUL THOUVENOT

ABSTRACT. We show that for any irrational numberαand a se- quence of integers{ml}_l∈Nsuch that lim

l→∞|||mlα|||=0, there exists a continuous measureµon the circle such that lim

l→∞ Z

T|||m_lθ|||dµ(θ) =0.

This implies that any rigidity sequence of any ergodic transfor- mation is a rigidity sequence for some weakly mixing dynamical system.

On the other hand, we show that for anyα ∈ _R−_{Q, there} exists a sequence of integers{m_l}_l∈N such that|||m_lα||| → 0 and m_lθ[1]is dense on the circle if and only ifθ∈/Qα+Q.

1. INTRODUCTION

We denoteT=_{R/Z. For}x∈ _Rwe denote by|||x||| :=infn∈_N|x− n|. We denote byx[1]the fractional part ofx.

In this note, we prove the following results.

Theorem 1. For anyα ∈ _R−_Q and a sequence of integers{m_l}_l∈N such that lim_l→_∞|||m_lα||| = 0, there exists a measure µ on T which has no atoms such that lim_l→∞R

T|||m_lθ|||dµ(θ) =0.

Theorem 2. For any α ∈ _R−Q, there exists a sequence of integers {m_l}_{l∈N such that}|||m_lα||| → 0 and such thatm_lθ[₁]is dense in Tif and only ifθ ∈/_Qα+_Q.

Due to the Gaussian measure space construction (see [4], or for ex- ample Proposition 2.30 in [2]), Theorem 1 has a direct consequence on rigidity sequences of weakly mixing dynamical systems. The fol- lowing is in fact an equivalent statement to Theorem 1.

Corollary 1. For any α ∈ _R−_Qand a sequence of integers{m_l}_l∈N such that lim_l→_∞|||m_lα||| = 0, there exists a weak mixing dynam- ical system (T,M,m) such that {m_l}_l∈N is a rigidity sequence for (T,M,m).

Date: January 7, 2014.

Supported by ANR-11-BS01-0004.

1

A consequence of Corollary 1 is a positive answer to a question raised in [2], namely : a rigidity sequence of any ergodic transformation (on a probability space without atoms) with discrete spectrum is a rigidity sequence for some weakly mixing dynamical system. Indeed, Corollary 1 deals with the case of a pure point spectrum with an irrational rota- tion of the circle as a factor. The case of a purely rational spectrum was treated in [2], Proposition 3.27. In case the spectrum is purely rational, our proof of Theorem 1 given below applies with only one modification: instead of working with the orbit of 0 under the rota- tion R_α, (α ∈/ Q/Z), one considers the union of the orbits of 0 under the actions of all the finite groups which appear in the (necessarily dense) support of the spectral measure.

A completely different solution to the same question was given in [1], who proved directly Corollary 1 based on a sophisticated and involved cut and stack construction.

In contrast, our proof is much simpler and is based on the straight- forward characterization of rigidity as a spectral property, which re- duces the answer to the question to the construction of a continuous probability measure on the circle with a Fourier transform converg- ing to 1 along the rigidity subsequence as stated in Theorem 1. This possible approach to the question was discussed in detail in [2].

The second result, Theorem 2, asserts that it is not possible to ex- pect more than what is obtained in Theorem 1, namely that a strong convergence of |||mnθ||| to 1 on an uncountable set K is not possi- ble in general for a sequence{mn}_n∈Nsuch that lim_l→_∞|||m_lα||| = 0, α ∈ _R−Q. Constructing such a setKwas a possible strategy to prov- ing Corollary 1 (see for example Proposition 3.3 in [2]), and Theorem 2 shows that this approach cannot be adopted in general.

Given an increasing sequence of integers {mn}_n∈_N, the study of the accumulation points on the circle of the sequence {mnξ}, for ξ irrational, has a long history and a rich literature (see for example [3, 5] and references therein). Weyl [6] proved that for any increasing sequence{_mn}_n∈N, it holds that for almost everyξ, {_mnξ} _{is dense} on the circle. The set of irrationalsξsuch that{mnξ}is not dense inT is called the set of exceptional points for the sequence{mn}_{n∈N. Our} result asserts the existence for anyα ∈_R−_Qof a sequence{mn}_n∈N for which the set of exceptional points is reduced to Qα+_{Q. To} our knowledge, no other examples of increasing sequences{mn}_n∈N with a countable exceptional set are known in the literature.

ACKNOWELEDGMENT. The authors are grateful to the referee for suggesting improvements to the first version of the paper.

ON THE CONVERGENCE TO 0 OF nξ 1 3

2. PROOF OF THEOREM1

Fixα ∈ _R−_Qand a sequence of integers{m_l}_{l∈Nsuch that}

(?) lim

l→_∞|||m_lα||| =0.

For a probability measureµonTwe writeµⁿ =|R

T|||mnθ|||dµ(θ)|. We will construct a sequence µp, p ≥ 0, of probability measures on Tof the form ₂¹p ∑²_i=^p1δx_i with x_i = k_iα such that there exists an increasing sequence{Np} for which

(1) For every p ≥ 1, for every j ∈ [0,p−1], for every n ∈ [N_j,N_j+1],µⁿ_p< ¹

2^j

(2) For every p₀ ∈_N^∗, if we let ηp₀ = ¹

4 inf

1≤i<i⁰≤2^p0|||k_iα−k_i⁰α|||

then for everyl ∈ _Nand everyr ∈[1, 2^p0],|||k_l2^p0+rα−krα||| <

ηp₀

(3) µⁿ_p < ¹

2^p+1 forn≥ Np.

When going from the measure µp toµp+1 we will add2^p masses at points selected nearbyx₁, . . . ,x₂^p that are already chosen forµp.

Theorem 1 clearly follows from the above construction. Indeed, property (1) will imply that any weak limitµ_∞ofµpsatisfiesµⁿ_∞ →0.

While by (2) we get that for each p0 the intervals (krα−ηp₀,krα+ ηp₀),r ∈ [1, 2^p0], on the circle are disjoint and each have mass 1/2^p0 for everyµp, p≥ p0, and hence forµ_∞that has therefore no atoms.

Property (3) is not necessary in the proof of the theorem, but it is useful to fulfill the inductive hypotheses (1) and (2) of the construc- tion.

For p=0, we letk1 =0 andµ0is thus the Dirac measure at 0. We letN0 =0. Forp =1, we letk2 =1 soµ1is the average of the Dirac measures at 0 and atα. Observe that for any n,µⁿ₁ < ¹₂ which fulfills (1) for p=1. We also chooseN1sufficiently large so thatµⁿ₁ < ₂¹₂ for n ≥N₁, the latter being possible due to(?).

We now assume selected k_i fori ≤ 2^p and N_l for l ≤ p such that (1) and (3) are satisfied up to p, and (2) is satisfied for every p0 ≤ p and every 0≤l ≤2^p−p0 −1.

We choosek₂^p+₁such thatk₂^p+₁αis sufficiently close tok₁αso that νp,1 = ¹

2^p+1

2^p+1 i

∑

δ_k⁰

iα

where k⁰_i = k_i for i ≤ 2^p and k⁰₂p+1 = k₂^p+1 while k⁰₂p+r = kr for r ∈ [2, 2^p]satisfiesνⁿ_p,1 < ¹

2^j for everyn∈ [N_j,N_j+1]andj∈ [0,p−1]. Since for every n we have that |_νⁿ_p,1−_µⁿ_p| < ¹

2^p+1|||mnk₂^p+1α− mnk₁α||| < ¹

2^p+1 we get by (3) that νⁿ_p,1 < ¹

2^p+1 + ¹

2^p+1 < ₂¹p for ev- eryn ≥ Np. Next we choose, N_p,1 > Np sufficiently large such that ν_p,1ⁿ < _2p+2¹ for n ≥ N_p,1, which is possible by (?). In this way, we select inductivelyk₂^p+s, then Np,s fors =1, 2, . . . , 2^p, and set

νp,s = ¹ 2^p+1

2^p+1 i

∑

δ_k⁰

iα

wherek⁰_i =k_ifori≤2^p+sandk⁰₂p+t = kt fort ∈ [s+1, 2^p]. Choos- ing for each s, k2^p+sα sufficiently close to ksα then Np,s sufficiently large we can insure that

• _νⁿ_p,s < ¹

2^j for everyn∈ [N_j,N_j+1]andj ≤ p−1.

• νⁿ_p,s < ₂¹p for everyn ≥Np

• νⁿ_p,s < _2p+2¹ for everyn ≥Np,s

The first point can be established inductively due to the fact that if k₂^p+sα is chosen very close toksα then the measuresν_p,s−1 and νp,s

are very close.

The same argument gives the second point for Np ≤ n ≤ N_p,s−1. As forn ≥ N_p,s−1 we use the fact that |_νⁿ_p,s−_νⁿ_p,s−1| < ¹

2^p+1 and the fact thatνⁿ_p,s−1 < _2p+2¹ for everyn≥ Np,s−1to conclude thatνⁿ_p,s < ₂¹p. For the third point we just choose Np,s sufficiently large and use(?). Finally, we let Np+₁ =Np,2^p andµp+₁ =νp,2^p and observe that the measureµp+1satisfies (1).

Also, since k₂^p+sα can be chosen arbitrarily close to ksα for s = 1, . . . , 2^p, we get that for every p₀ ≤ p+1, for every l = 2^p−p0 + l⁰−_1, l⁰ ≤ ₂^p−p0_, |||k_l2^p0+rα||| ∼ |||k_l⁰₂^p0+rα||| ∼ |||krα||| from where (2) follows for p+1. The proof of Theorem 1 is thus complete.

3. PROOF OF THEOREM2 In all this sectionα∈ _R−_Qis fixed.

Definition1. For an interval I ⊂_{T, forε} > 0, and integers N₁ < N₂, we say thatθ ∈ A(N₁,N₂,I,ε,α) if for everym ∈ [N₁,N₂)such that

|||mα||| <εwe have that{mθ} ∈/ I.

Lemma1. For everyl ≥2, there existsL(l) ∈ N, such that for every 0 < ε ≤ _2l¹₂, for everyν > 0, N ∈ N, there exist K(ε) >0 and N⁰ =

ON THE CONVERGENCE TO 0 OF nξ 1 5

N⁰(l,ε,ν,N) ∈ _Nsuch that θ ∈ A(N,N⁰,I,ε,α)for some interval I of size 1/l, implies that|||kα−sθ||| <νfor some|k| ≤K(ε)and some

|s| ≤ L(l).

Proof. For any ε > 0, consider an approximation by trigonometric polynomials of 2χε whereχεis the characteristic function of the sub- set ofT[0,ε]∪[1−ε, 1], namelyφε :T→_Rsuch that

• φε(x)>1 for everyx ∈ [0,ε]∪[1−ε, 1]

• _φε(x)>−_ε³for everyx ∈_T

• There existsK ∈_Nsuch thatφε(x) =_∑|k|≤Kφ^ˆ_ke^i2πkx. Similarly, forl ≥2, letϕl :T→_Rbe such that

• _ϕl(y)>1 for everyy∈/[0,¹_l]

• |ϕ_l(y)| <l²for everyy∈ _T

• There exists L∈ _Nsuch that ϕ_l(y) = _∑0<|k|≤L ϕˆ_ke^i2πky. Note that the second requirement includes the fact thatR

ϕ_l(y)dy = 0.

Forψ :T² →_Rand (α,θ)∈ _R²we define fork∈ _N S^α,θ_k ψ(x,y) =

k−1 i

∑

ψ(x+iα,y+iθ). Fix I = [y0,y0+¹_l], for somey0∈ _T,l ≥2.

Defineψ_ε,l : T² → _Rbyψ_ε,l(x,y) = φε(x)ϕ_l(y−y0). For N⁰ ∈ _N sufficiently large there exist more thanε²N⁰integersi ∈[N,N⁰)such that |||iα||| < ε. If θ ∈ A(N,N⁰,I,ε,α) then S^α,θ_N0ψ_ε,l(0, 0) > (ε²− l²ε³)N⁰ ≥ ¹₂ε²N⁰.

On the other hand, we have that S^α,θ_N0ψ_ε,l(x,y) =

∑

|k|≤K,0<|j|<L

φˆ_kϕˆ_j1−e^{i2πN0(kα+jθ)}

1−e^{i2π(kα+jθ)} e^i2π(kx+jy) hence, if |||kα−jθ||| ≥ _ν for every |k| ≤ K and every 0 < |j| ≤ L, thenS^α,θ_N0ψ_ε,l(x,y)is bounded independently ofN⁰which contradicts S^α,θ_N0ψ_ε,l(0, 0)> ¹₂ε²N⁰.

Proof of Theorem 2. Define for n ≥ 1, the sequenceln = n+1 and Ln := L(ln) given by Lemma 1. Let εn = _2(n+¹₁₎₂ and Kn = K(_εn) given by Lemma 1. Letνn = ¹_{n inf0<|k|≤(n+1)K}

n+1|||kα|||_{. Take}N₀ = ₀ and apply Lemma 1 withl =l1,ε =ε1, N = N0andν =ν1. Denote then N1 = N⁰(l1,ε1,ν1,N0). We then apply inductively Lemma 1

withl = ln, ε =εn, N = Nn and ν =νn and choose N_n+1arbitrarily large such that Nn+₁ ≥N⁰(ln,εn,νn,Nn).

We define an increasing sequence m_l by taking successively for everyi, all the integersm∈ [Ni,Ni+₁)such that|||mα||| <εi(choosing Nn+1to be sufficiently large in our inductive construction guarantees that the sequencemnis not empty).

Suppose nowθis such thatmnθ[1]is not dense on the circle. Then, there existskand an intervalIof sizelksuch that for everyn,mnθ[1]∈/ I. In other words,θ ∈ A(Nn,N_n+1,I,εn,α) _{for every}n ≥ n₀, n₀ suf- ficiently large. LetL =L_k. By Lemma 1 we get that|||knα−lθ||| <_νn for some|kn| ≤ Kn and some 0<|l| ≤ L. Hence|||k⁰_nα−L!θ||| <L!νn

for some|k⁰_n| ≤ L!Kn. It follows that|||(k⁰_n+1−k⁰_n)_α||| <2L!νn. From the definition ofνn this implies that k⁰_n+1 = k⁰_n for sufficiently large n, say n ≥ n1. Since νn → 0, we get that|||k⁰_n1α−L!θ||| = 0, which givesθ ∈ _Qα+Q. Conversely, {mnθ} forθ ∈ _Qα+_Qis clearly not dense on the circle and Theorem 2 is proved.

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IMJ-PRG, CNRS, UP7D-CAMPUSGRANDMOULIN, B ˆATIMENTSOPHIEGER-

MAIN, CASE7012, 75205 PARISCEDEX13,BASSAM@MATH.JUSSIEU.FR

LPMA, UNIVERSITE´ PIERRE ET MARIE CURIE, 4 PL JUSSIEU, 75252 PARIS,

JEAN-PAUL.THOUVENOT@UPMC.FR