Almost everywhere convergence of convolution powers on compact abelian groups

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www.imstat.org/aihp 2013, Vol. 49, No. 2, 550–568

DOI:10.1214/11-AIHP468

Almost everywhere convergence of convolution powers on compact Abelian groups

Jean-Pierre Conze

^a

and Michael Lin

^b

aIRMAR, CNRS UMR 6625 Université de Rennes 1, 35042 Rennes Cedex, France. E-mail:[email protected] bDepartment of Mathematics, Ben-Gurion University, Beer-Sheva, Israel. E-mail:[email protected]

Received 24 August 2011; revised 22 November 2011; accepted 22 November 2011

Abstract. It is well-known that a probability measureμon the circleTsatisfiesμⁿ∗f −

fdmp→0 for everyf ∈Lp, every (some)p∈ [1,∞), if and only if| ˆμ(n)|<1 for every non-zeron∈Z(μis strictly aperiodic). In this paper we study the a.e.

convergence ofμⁿ∗f for everyf∈L_pwheneverp >1. We prove a necessary and sufficient condition, in terms of the Fourier–

Stieltjes coefficients ofμ, for the strong sweeping out property (existence of a Borel setB with lim supμⁿ∗1_B=1 a.e. and lim infμⁿ∗1_B=0 a.e.). The results are extended to general compact Abelian groupsGwith Haar measurem, and as a corollary we obtain the dichotomy: forμstrictly aperiodic, eitherμⁿ∗f→

fdma.e. for everyp >1 and everyf∈Lp(G, m), orμhas the strong sweeping out property.

Résumé. Il est connu qu’une mesure de probabilitéμ sur le cercleTsatisfaitμⁿ∗f −

fdmp→0 pour toute fonction f∈L_pet pour toutp∈ [1,∞)(ou pour unp∈ [1,∞)), si et seulement siμest strictement apériodique (i.e.| ˆμ(n)|<1 pour tout nnon nul dansZ). Nous étudions ici la convergence presque partout deμⁿ∗f pourf∈L_p,p >1. Nous montrons une condition nécessaire et suffisante portant sur les coefficients de Fourier–Stieltjes deμpour la propriété de “balayage fort” (existence d’un borélienBtel que lim supμⁿ∗1_B=1 p.p. et lim infμⁿ∗1_B=0 p.p.). Les résultats sont étendus aux groupes abéliens compacts générauxGde mesure de Haarm. Comme corollaire nous obtenons la dichotomie suivante : pourμstrictement apériodique, soit μⁿ∗f→

fdmp.p. pour toutp >1 et toute fonctionf∈L_p(G, m), soitμvérifie la propriété de balayage fort.

MSC:Primary 37A30; 28D05; secondary 47A35; 60G50; 42A38

Keywords:Convolution powers; Almost everywhere convergence; Sweeping out; Strictly aperiodic probabilities

1. Introduction

Let (X,B)be a measurable space and P (x, A):X×B−→ [0,1] a transition probability, with Markov operator Pf (x)=

f (y)P (x,dy)defined for boundedf. Whenmis a probability onBwhich isP-invariant, the operatorP can be extended to a contraction ofL₁(X, m). MoreoverP becomes a contraction in eachL_p(X, m)space, 1≤p≤ ∞ [13].

Hopf’s pointwise ergodic theorem yields that forf ∈L1(m)the Cesàro averages¹_nn

k=1P^kf converge a.e. and in L_p-norm whenf ∈L_p(m), 1≤p <∞. The limit is

fdmifP is ergodic inL₁, i.e. whenPf =f a.e. forf ∈L₁ holds only forf constant a.e.

It is therefore a natural question to study the convergence of the unaveraged sequence{Pⁿf}, in norm or a.e. The following general results for a.e. convergence are known:

1. IfP^∗=P and−1 is not an eigenvalue, thenPⁿf converges a.e. for everyf ∈L_p,p >1 (Stein–Rota theorem [23,25]; Rota’s proof yields the convergence also forf ∈Llog⁺L[6], but in general convergence may fail forf ∈L₁ [17]).

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2. IfP is an aperiodic Harris recurrent operator, thenPⁿf →

fdma.e. for everyf∈L₁(X, m)(mis assumed finite), by S. Horowitz [11].

Often, a.e. convergence of {Pⁿf}n≥1 for every bounded measurable function fails in a very strong manner ex- pressed in the following definition, which seems to have been introduced for operator sequences{Pn}in the study of a.e. convergence of averages along subsequences.

Definition 1.1. We say that{Pⁿ}(or simplyP) has the strong sweeping out property (SSO property)if there exists in Ba denseG_δsubset of setsB of positive measure such thatlim sup_nμⁿ∗1B=1a.e.andlim infnμⁿ∗1B=0a.e.

In this paper we study the strong sweeping out property for the convolution operator P_μ defined by a strictly aperiodic probability measure μ on a compact Abelian group G, P_μf (x)=μ∗f (x)=

Gf (x+y)dμ(y). We obtain a necessary and sufficient condition for the strong sweeping out property forPμ(we will say simply “forμ”), in terms of the Fourier–Stieltjes transform ofμ, and deduce the dichotomy:eitherμⁿ∗f →

fdma.e. for every f ∈Lp(G, m), everyp >1, orμhas the strong sweeping out property.

For the sake of clarity, we prove the results first for convolutions on the unit circle (Section 3), and after some examples for discrete probabilities on the circle (Section4), we add the necessary ingredients to prove the result in the general case (Section5).

2. Convolution powers on compact Abelian groups

In this section we look at the problem of almost everywhere convergence (to the integral) of convolution powers of a probability μ on a compact Abelian groupG, with Borel σ-algebraB and dual groupG. Characters onˆ Gwill be denoted by γ. The Markov transition is P (x, A)=μ(A−x), with invariant probability the normalized Haar measurem, and the corresponding Markov operator isPf:=P_μf =μ∗f. The dual Markov operator isP_μ^∗=P_μ_ˇ, whereμˇ is the reflected probability given byμ(A)ˇ =μ(−A). By commutativity ofG, the operatorP_μis normal in L₂(G, m). We note that the Markov chain{Y_n}onΩ=G^N induced byP_μis the random walk onGof lawμ, and μⁿ∗1A(x)=Px{Y_n∈A}, wherePxis the probability onΩ for the chain started atx(initial distributionδ_x).

The Fourier–Stieltjes coefficientsμ(γ )ˆ are eigenvalues ofP_μwith continuous eigenfunctions, so a necessary condition for a.e. convergence of{μⁿ∗f}to the integral for all continuous functions is that| ˆμ(γ )|<1 for everyγ =0, i.e.μisstrictly aperiodic. We recall the well-known properties equivalent to strict aperiodicity of a probabilityμon a compact Abelian groupG:

Proposition 2.1. The following are equivalent:

(i) | ˆμ(γ )|<1for every character0 =γ∈ ˆG;

(ii) μⁿ∗f →

Gfdmuniformly for every continuous function onG;

(iii) the support ofμis not contained in a class of a proper closed subgroup;

(iv) μⁿ∗f−

Gfdm2→0for everyf ∈L₂(G, m).

It follows thatμⁿ∗f →

fdmpointwise forf in a dense subspace ofL_p, 1≤p <∞, namelyC(G). However, a result of J. Rosenblatt [20] yields that forα=e²^π^iθ withθ∈(0,1)irrational, the strictly aperiodicμ=¹₂(δ1+δα) is strongly sweeping out onT.

Some of the general results of a.e. convergence cited in theIntroductioncan be improved for the powers of the convolution operatorP_μin several particular cases:

1. Ifμis symmetric and strictly aperiodic, then for everyf ∈LlogLwe haveμⁿ∗f →

fdma.e. [23] (see also [18]).

2. In any compact groupG(not necessarily Abelian), if some powerμ^k is non-singular with respect to the Haar measurem, thenμⁿ−m →0 in total variation norm with exponential rate (cf. [2], Theorem 3, forGconnected, [1], Theorem 4.1, forGnot necessarily connected and for the precise rate); see [22], Theorem 4.1, for a list of equivalent conditions. In this case, for everyf inL1(G, m)the series_∞

n=1μⁿ∗(f−

fdm)converges a.e. It follows that any μwith the strong sweeping out property has all its convolution powers singular (cf. Proposition3.4).

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3. Another sufficient condition for a.e. convergence is sup_γ₌₀| ˆμ(γ )|<1. It implies sup_f_p_≤₁μⁿ ∗f − fdmp→0 (exponentially fast) forp∈(1,∞)by [21], p. 202ff (see also [9], Proposition 4.1); so forp∈(1,∞), the series_∞

n=1μⁿ∗(f−

fdm)converges a.e. for anyf ∈L_p(G, m).

In particular, on the circleT, the condition supγ =0| ˆμ(γ )|<1 holds whenμ(n)ˆ →0 as|n| → ∞[9]. Note that the above norm convergence needs not hold forp=1 orp= ∞[9].

4. There exists onTa continuous probabilityμwith all its convolution powers singular, such that sup_n₌₀| ˆμ(n)|<

1 (and then, by [21],μⁿ∗f→

fdma.e. for everyf ∈L_p(T, m),p >1). See [9], Proposition 4.7, for a construc- tion along classical lines. A result of Varopoulos [26] shows that we can findμcontinuous with all its convolution powers singular satisfyingμ(n)ˆ →0 as|n| → ∞(see also the first lines following [19], Theorem 4.2). (The result of Varopoulos is that in any non-discrete compact Abelian group there is a probabilityμwith all its convolution powers singular to the Haar measure such thatμ(γ )ˆ vanishes at infinity.)

Notations. The spectrum of the operatorPf =μ∗f inL₂(T, m)is denoted byσ (P ).It is the closure of{ ˆμ(γ ): γ∈ Gˆ}.In the sequel we study the peripheral spectrum ofP,i.e.the unimodular complex numbers inσ (P ).It will be useful to distinguish notationallyT,the group on which the convolution operates(the state space of the Markov chain generated),fromS=S¹,the boundary of the closed unit disk which contains the spectrum.

Since the spectral radius of an operator is bounded by its norm, and using the spectral theorem, one easily proves the equivalence between the following conditions:

(i)Pⁿ(I−P )

2→0, (ii) sup

λ∈σ (P )

λⁿ(1−λ)→0, (1) (iii) sup

γ∈ ˆG

μˆⁿ(γ )

1− ˆμ(γ )→0, (iv)σ (P )∩S= {1}. (2)

Remarks. 1.Ifμⁿ⁺¹−μⁿ1→0,then for1≤p <∞continuity of the representation ofGinL_p(G, m)by trans- lations yieldsPⁿ⁺¹−Pⁿp→0.

2.IfPⁿ⁺¹−Pⁿ2→0,then by the Riesz–Thorin theoremPⁿ⁺¹−Pⁿp→0for every1< p <∞.However, forp=1we may still haveμⁿ⁺¹−μⁿ1=2for everyn(e.g., [12],Remark2.16(b)).

Proposition 2.2. Letμbe a strictly aperiodic probability on a compact Abelian groupG.Ifσ (P )∩S =S,then there exists an increasing subsequence{n_k}such thatμⁿ^k∗f →

fdma.e.for everyf ∈L₂(G, m).

Proof. SinceP is a positive contraction, the assumptionσ (P )∩S =Simplies that this intersection is finite, and for somej≥1 we haveσ (P^j)∩S= {1}[14], Proposition 1 (see Lemma3.5below for a direct proof). Therefore (iii) of (2) holds forμ^j and it follows from [12], Proposition 2.15, that there exists an increasing subsequence{ k}such that μ^j^k∗f →

fdma.e. for everyf ∈L2(G, m).

Remark. The probabilityμ=¹₂(δ₁+δ_α)onThas the strong sweeping out property as mentioned above,although σ (P_μ)∩S= {1}(and alsoμⁿ⁺¹−μⁿ1→0by Foguel’s zero-two law).By Proposition2.2there is a subsequence {n_k}withμⁿ^k∗f →

fdmfor everyf∈L2(T, m).

Now we give a variant of a result of [5] (see also [12], Theorem 2.20) which gives a sufficient condition for the a.e. convergence forf inL_p,p >1. It is based on Theorem 14 of [4], or the following extension of it which does not require normality.

We consider a positive contractionT onL₂(X, m)where(X, m)is a probability space. For every integerr∈ [1, n], letΔ^rTⁿ:=Tⁿ⁻^r(T −I )^r, whereI is the identity onL₂(X, m).

Theorem 2.3. Let T be a positive contraction of L2(X, m),withW be a closedT-invariant subspace.Let W =

j∈JV_j be an orthogonal decomposition ofW into closedT-invariant subspaces such that the restrictionT_j ofT toV_j satisfiesT_j<1.LetL(j ):=₁^I₋^j⁻_T^T^j

j,whereI_j is the identity onV_j.

Putf₀^∗(x):=sup_n_≥₀|Tⁿf (x)|andf_r^∗(x)=sup_n_≥_r|n^rΔ^rTⁿf (x)|,forr≥1.Then:

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(i) Forf∈W,iff =

jΠ_jf is its orthogonal decomposition,the maximal function satisfies f₀^∗

2≤2f2+

j

Π_jf²₂L(j )² 1/2

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and

jΠjf²L(j )²<∞implies the convergencelimnTⁿf=0m-a.e.and inL2(m).

(ii) IfL_∞:=sup_jL(j ) <∞,there are finite constantsC_r such that for everyf∈W, f_r^∗

2≤C_rf2, ∀r≥0, (4)

and for everyr≥0,the convergencelimnn^rΔ^rTⁿf =0holdsm-a.e.and inL2(m).

Proof. (i) The proof is based on the idea of comparingTⁿwith its Cesàro averages, like the proof of Stein’s theorem for self-adjoint positive contractions [25] or that of [4] for positive normal contractions, but does not require the spectral theorem.

Forn≥1 put A_nf =¹_n_n₋₁

k=0T^kf. By Akcoglu’s ergodic theorem [13], p. 190, for everyf ∈L₂the sequence Anf (x)converges a.e. and the maximal functionA^∗f (x)=sup_k_≥₀|Akf|(x)satisfies A^∗f2≤2f2. Using the inequality

1 n

n

k=1

ka_k

2

≤ n

k=1

k|a_k|², (5)

which holds for every complex numbersa₁, . . . , a_n, we can write forf ∈W: Tⁿf−A_nf²=

1 n

n

k=1

k

T^kf−T^k⁻¹f

2

≤ n

k=1

kT^kf −T^k⁻¹f²≤F₁(f )², withF₁(f )²=_∞

k=1k|T^k⁻¹(T −I )f|².

Observe that for 0≤λ <1, we have(1−λ)²_∞

1 kλ^2(k⁻¹⁾=1/(1+λ)²≤1. Therefore, F₁(f )²

2=^∞

k=1

k

X

j

T_j^k⁻¹(T_j−I_j)Π_jf

²dm=

k

j

T_j^k⁻¹(T_j−I_j)Π_jf²

2

≤

j

k

kTj^2(k⁻¹⁾

Tj−Ij²Πjf²2≤

j

T_j−I_j²

(1− T_j)²Πjf²2=

j

Πjf²2L(j )².

This proves (3), since we have|Tⁿf| ≤A^∗f +F₁(f ),∀n≥1; hencef₀^∗≤A^∗f +F₁(f ). The convergence in L²-norm follows from the assumptionT_j<1.

LetKbe a finite subset ofJ. Writef =ϕK+ρK, withϕK=

j∈KΠjf andρK=

j /∈KΠjf. Forε >0, take Ksuch thatρ_K2≤εand

j /∈KΠ_jf²₂L(j )²≤ε².

lim sup

n

Tⁿf

2≤(ρ_K)^∗₀

2≤A^∗ρ_K

2+F₁(ρ_K)≤2ρ_K2+

j /∈K

Π_jf²₂L(j )² 1/2

≤3ε.

(ii) Forr≥1,x∈ [0,1[, we have_∞

k=rk^2r⁻¹x^k⁻^r=ρ_r(x)/(1−x)^2r, whereρ_r is a polynomial. Therefore there exists a finite constantD_r such that

(1−λ)^2r ∞ k=r

k^2r⁻¹λ^2(k⁻^r)=ρr

λ²

/(1+λ)^2r≤D²_r, 0≤λ <1. (6)

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The relationC^r_k⁻¹+C_k^r=C_k^r₊₁, satisfied by the binomial coefficientsC_k^r, yields C_n^r⁺₊¹₁Δ^rTⁿ=

n

k=r

C_k^rΔ^rT^k+ n

k=r+1

C_k^r⁺¹Δ^r⁺¹T^k, 0≤r≤n. (7)

Forf ∈L₂(X, m), it follows from (5):

1 n

n

k=r

C_k^rΔ^rT^kf

2

≤ n

k=r

k

C^r_k/k2T^k⁻^r(T −I )^rf²≤F_r²,

with F_r=

_∞

k=r

k⁻¹

C_k^r2T^k⁻^r(T−I )^rf²1/2

.

From the decomposition in the orthogonal subspacesV_j, we get:

F_r²2≤

j

_∞

k=r

k⁻¹ C_k^r2

T_j^2(k⁻^r)

T_j−I_j^2rΠ_jf². (8)

According to (6) withλ= T_j, the series in[ ]in (8) is less thanD_r²/(1− T_j)^2r, so F_r²₂≤D_r²

j

L(j )^2rΠ_jf²≤D_r²

sup

j

L(j ) 2r

j

Π_jf²=D²_rL^2r_∞f².

Forn≥r≥1 the identity (7) implies the inequality 1

n+1C^r_n⁺₊¹₁Δ^rTⁿf ≤ 1

n+1

n

k=r+1

C_k^r⁺¹Δ^r⁺¹T^kf + 1

n+1

n

k=r

C_k^rΔ^rT^kf

≤F_r₊₁+F_r;

sinceC_n^r⁺₊¹₁=ⁿ_r⁺₊¹₁_r₋₁

j=0 n−j

r−j ≥ⁿ_r⁺₊¹₁(ⁿ_r)^r forn≥r≥1, puttingB_r =(r+1)r^r we obtain f_r^∗

2=sup

n≥r

n^rΔ^rTⁿf

2≤Br

sup

n≥r

1

n+1C_n^r⁺₊¹₁Δ^rTⁿf

2

≤Br

Fr+12+ Fr2

≤Br

Dr+1L^r_∞⁺¹+DrL^r_∞ f2.

The proof of the convergence statement in (ii) is similar to that of (i).

Suppose now thatT defines a positive contraction of eachLp(m), 1≤p≤ ∞, andW =L2(m)in Theorem2.3.

The inequalities (4) and the classical inequalitiessup_n_n₊¹₁(f+Tf + · · · +Tⁿf )p≤_p^p₋₁fp, f ∈L_p(m), 1<

p <∞, are the needed properties for Stein’s complex interpolation theorem [25] (see also [7] for a detailed presenta- tion of the method applied to iterates of composed conditional expectations). It implies for 1< p <∞the maximal inequality and the a.e. convergence ofTⁿf forf ∈L_p(m). This applies to convolution powers, i.e.T =P_μ, even on non-Abelian (compact) groups. In the Abelian case, it yields:

Theorem 2.4 ([4]). Letμbe a strictly aperiodic probability on a compact Abelian groupG.If

sup

γ =0

|1− ˆμ(γ )|

1− | ˆμ(γ )|<∞, (9)

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then forp >1there is a constantC_p such that for everyf ∈L_p(G, m),the maximal inequalityf₀^∗p≤C_pfp

holds andμⁿ∗f→

fdma.e.

Proof. LetPf =μ∗f be the normal operator induced onL₂(G, m). Thenσ (P )is the closure of{ ˆμ(γ ): γ∈ ˆG}, and (9) implies thatσ (P )contains no unimodular points except 1 and is included in a Stolz region of the closed unit disk. Using part (ii) of Theorem2.3, whenVγ is the space of multiples of the characterγ∈ ˆG, the maximal inequality

and the a.e. convergence follow from Stein’s theorem as explained above.

As mentioned in theIntroduction, failure of a.e. convergence for a sequence of operators may be quite strong. In [8], del Junco and Rosenblatt gave a condition which implies the SSO property (cf. Definition1.1) and, for powers of convolution operators onG, reads:

Theorem 2.5. Letμbe a probability measure on the compact Abelian groupG.If for every integerN0>1and every ε >0,there exists a measurable setAsuch thatm(A) < εandm{sup_n_≥_N₀μⁿ∗1A(x)≥1−ε}>1−ε,thenμhas the strong sweeping out property.

The next lemma shows that it suffices to fulfill the conditions forN₀=1.

Lemma 2.6. If for everyε >0there exists a setA∈Bsuch thatm(A) < εand m

x∈G: sup

n≥1

μⁿ∗1A(x)≥1−ε

>1−ε, (10)

thenμhas the strong sweeping out property onG.

Proof. Let be given N₀≥1 and ε >0. By applying (10) with ε= _2N^ε₀₊_ε instead ofε, we obtain a setA with μ(A) < εwhich satisfies the condition of Theorem2.5since

m

sup

n≥N0

μⁿ∗1A(x)≥1−ε ≥m

sup

n≥N0

≥m

sup

n

−N0m(A)/

1−ε

≥1−ε−N0m(A)/

1−ε

≥1−ε−N0ε/ 1−ε

≥1−ε.

3. Convergence and divergence of convolution powers on the circle

Our aim is to characterize the strong sweeping out property on the circleT by a property of the Fourier–Stieltjes coefficients of μ, and to study the a.e. convergence of the convolution powers of μ. In order to avoid repetition when dealing with general groups, we denote the characters byγ and the Fourier–Stieltjes coefficients byμ(γ ). Forˆ γ (x)=e²^π^inx, we write eitherμ(γ )ˆ orμ(n).ˆ

Theorem 3.1. Letμbe a strictly aperiodic probability on the unit circleT.If lim sup

ˆ

μ(n)→1,n =0

|1− ˆμ(n)|

1− | ˆμ(n)|= ∞, (11)

thenμhas the strong sweeping out property onT.

Proof. We will use Lemma2.6to show the SSO property. For a characterγ =0 denote L(γ )=|1− ˆμ(γ )|

1− | ˆμ(γ )| and ρ(γ )=

1− μ(γ )ˆ

| ˆμ(γ )|

. (12)

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L(γ )is well-defined by strict aperiodicity,L(γ )≥1 and the triangle inequality yields ρ(γ )≥1− ˆμ(γ )−

μ(γ )ˆ − μ(γ )ˆ

| ˆμ(γ )| =

1−μ(γ )ˆ L(γ )−1

. (13)

By (11), there exists an infinite sequence{γk},γk(x)=e²^πⁱⁿ^k^xforx∈R/Z, such thatL(γk)→ ∞andρ(γk)→0.

Let 0< ε <1 be given and putM=¹_2ε^+π. We will constructA∈Bsatisfying (10), by adapting the ideas of Losert [16]. From the above sequence{γk}, we fixγksuch that

ρ(γ_k) < 1

M and L(γ_k) >32M²π

ε . (14)

ρ L ≥

1−μ(γˆ k)L−1 L ≥1

2

1−μ(γˆ k). (15)

Putj:= [²_ρ^π] +1 andr:=1− | ˆμ(γk)|^j. By (15) we have 1−r≥(1−^2ρ_L)^j>0. Then

−r≥ln(1−r)≥jln

1−2ρ L

≥ −j4ρ

L, (16)

since ln(1−t )≥ −2t for t <1/4, while ρ <1/M and L >32π imply _L^ρ ≤1/32π. The estimate (16) and the definition ofj yield

r≤4ρj L ≤16π

L . (17)

We now defineδ:=max{^ρ₄, r

2ε}. The estimates (14) and (17) yield r

2ε≤ 8π

Lε<

8π 32M²π= 1

2M, (18)

which showsδ <1/2Msinceρ <1/M.

Since 2|sin(πξ )| =ρ <1, we haveξ <1/6. We saw thatρ≤2πξ, so 1

j < ρ

2π< ξ <1

2sin(πξ )=ρ

4 < δ. (19)

Hence thej intervals[( −1)ξ, ξ )mod 1,1≤ ≤j, each of lengthξ >1/j, cover all the unit interval, so for each x∈ [0,1)there exists xwith|n_kxmod 1− xξ|< ξ < δ(recall thatγ_k(x)=e²^πⁱⁿ^k^x).

Fix 1≤ ≤j. Using the definition ofξwe obtain μˆ (nk)=μ(nˆ k) =e⁻²^π^{i ξ}

ˆ μ(nk)

=e⁻²^π^{i ξ}

I

e⁻²^πⁱⁿ^k^sdμ (s)

=

I

e⁻²^πⁱ⁽ⁿ^k^s⁺^{ξ )}dμ (s)=

I

cos

2π(n_ks+ ξ )

dμ (s)=1−2

I

sin

π(n_ks+ ξ )2

dμ (s).

But for ≤j, we have| ˆμ(n_k)| ≥ | ˆμ(n_k)|^j=1−r, so

I

sin

π(nks+ ξ )2

dμ (s)=1 2

1−μ(nˆ k)

≤r 2.

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From Tchebyshev’s inequality we obtainμ {s∈I:|sin(π(n_ks+ ξ ))| ≥δ} ≤_2δ^r2. Givenx∈I, we have x≤j with

|n_kxmod 1− xξ|< δ. Hence|sin(π(n_ks+ xξ ))|< δimplies sin

π(n_ks+n_kx)≤sin

π(n_ks+ xξ )+sin

π(n_kx− xξ )< (1+π)δ.

Sinceδ²≥r/2εby the definition ofδ, it follows that μ ^x

s∈I:sin

π(n_ks+n_kx)< (1+π)δ

≥1− r

2δ²≥1−ε.

Defineϕ:I−→I byϕ(t )=n_ktmod 1. Thenϕpreserves Lebesgue’s measure onI. LetA=ϕ⁻¹(B), whereB= {s∈I: |sin(πn_ks)|< (1+π)δ}. Since|sin(πy)| ≥ |2y|for|y| ≤¹₂, we havem(A)=m(B)≤(1+π)δ <¹_2M^+π=ε and by the previous estimate

sup

1≤≤j

μ ∗1A(x)= sup

1≤≤j

μ (A−x)≥μ ^x(A−x)

=μ ^x

s∈I: sin

π(n_ks+n_kx)< (1+π)δ

≥1−ε.

This yieldsm{x∈ [0,1): sup_n_≥₀μⁿ∗1A(x) >1−ε} =m([0,1))=1, so (10) is satisfied.

Corollary 3.2. Let μbe strictly aperiodic such thatS⊂ { ˆμ(n): n∈Z}.Then(11)holds,and thereforeμ has the strong sweeping out property on the circle.

Proof. LetPf =μ∗f onL2(T, m)with spectrumσ (P ); it is easy to show thatσ (P )is the closure of{ ˆμ(n): n∈Z}.

Callnk a value of nk,j withj large so|1− ˆμ(nk,j)|<2|1−λk|andL(nk,j) > k. Thus (11) holds, and the strong

sweeping out property follows from the theorem.

Lemma 3.3 ([15], Lemma 1). Condition(11)is equivalent to lim sup

ˆ

μ(n)→1,n =0

|mμ(n)ˆ |

1− eμ(n)ˆ = ∞. (20)

Proof. For the sake of completeness, we give a proof and show the following equivalence: for every sequence{z_n} with|z_n|<1,

lim sup

|z_n|<1,z_n→1

|1−zn|

1− |zn|= ∞ if and only if lim sup

|z_n|<1,z_n→1

|mzn| 1− ezn = ∞.

Letz=ρe^iαbe a complex number with argumentα∈(−¹₂π,¹₂π)and modulusρ <1. PutA(z)=A(α, ρ)=^|₁¹_−|⁻^z_z^|_| andB(z)=B(α, ρ)=₁^|₋^mz_ez^|. We have

A(α, ρ)=

1+4ρ

sin(α/2) 1−ρ

21/2

, B(α, ρ)=2ρ cosα

2

|sin(α/2)| 1−ρcosα. PuttingM(z)=Mα,ρ=2^|^sin(α/2)₁₋_ρ ^|, we get

A(α, ρ)=

1+ρM_α,ρ² 1/2

, B(α, ρ)= ρ|cos(α/2)|

|sin(α/2)| +cosαM_α,ρ⁻¹.

Hence the equivalence between limz_n→1A(z_n)= ∞,limz_n→1M(z_n)= ∞,limz_n→1B(z_n)= ∞.

(9)

Remarks. 1.Even forμstrictly aperiodic,the conditionsup_n₌₀^|₁¹_{−| ˆ}^{− ˆ}^μ(n)_μ(n)^|_| = ∞is insufficient for the strong sweeping out(and(9)is not necessary for a.e.convergence).Letθ∈(0,1)be irrational,α=e²^π^iθ ∈T,and putμ=¹₂(δ_α−1+ δ_α).The operatorPf =μ∗f is self-adjoint on L₂(m),so by the Stein–Rota theorem sup_nPⁿ|f| ∈L₂ for every f∈L₂.We haveμ(n)ˆ =cos(2πnθ ),soμis strictly aperiodic,henceμⁿ∗f→

fdμa.e.for everyf∈L₂,and thus μdoes not have the sweeping out property. (Note that the result of[5]does not apply toμ,because¹₂(δ₁+δ₋₁)is not strictly aperiodic onZ,but it applies toμ².)We havesup_n₌₀^|₁¹_{−| ˆ}^{− ˆ}^μ(n)_μ(n)^|_|= ∞sinceL(n_k)→ ∞whenn_kθ→1/2 mod 1, but(11)fails since whenμ(n)ˆ is close to1its values are positive reals andL(n)=1.

2.Letα=e²^π^iθ be as above,and defineμ=¹₂(δ₁+δ_α).Thenμ(n)ˆ =¹₂(1+α⁻ⁿ),soμis strictly aperiodic.For z=¹₂(1+e⁻²^π^inθ),we have|z| = |cos(πnθ )|and|1−z| = |sin(πnθ )|,soL(n)→ ∞asnθ→0 mod 1.Hence(11) holds.

3.Letα=e²^π^iθ be as above.Theorem3.1forμ=

k∈Zp_kδ_αk (wherep_k≥0with

k∈Zp_k=1)follows from Theorem2of[16]:We put onZthe probabilityν:=

k∈Zp_kδ_kand obtainμ(n)ˆ = ˆν({nθ}).Henceμ(n)ˆ →1implies {nθ} →0,so(11)implieslim sup_t_→₀^|₁¹_−|ˆ^−ˆ^{ν(t )}_{ν(t )}^|_|= ∞.

Example 1. Let{α_k=e²^π^iθ^k}^d_k₌₁⊂Twithd >1,and assume that1, θ1, θ₂, . . . , θ_dare linearly independent over the rationals.Letμ=_d

k=1p_kδ_α_k (where0≤p_k<1and

p_k=1).Thenμhas the strong sweeping out property onT. Proof. We haveμ(n)ˆ =d

k=1pkα_k⁻ⁿ, andμis strictly aperiodic since its support has at least two “irrational” points.

The linear independence implies, by a result of Kronecker (e.g. [10], p. 382, [13], pp. 12–13), that the powers of (α₁, . . . , α_d)are dense in the d-dimensional torusT^d; hence for λ∈S there exists a subsequence{n_j}such that α_kⁿ^j →λ for k=1, . . . , d, which yields thatμ(nˆ j)→λ. We conclude thatS⊂ { ˆμ(n): n∈Z}, and Corollary3.2

applies.

Remark. Since the flow defined on(T, m)byT_tx=e²^π^itx is periodic,ford=3we cannot obtain Example1from Theorem2.18of[12] (which yields only non-convergence for somef ∈L₂(T, m),and not the strong sweeping out property).

Proposition 3.4. There exists a continuous probabilityμonTwhich has the strong sweeping out property onT. Proof. Recall that a closed set K⊂Tis called aKronecker set if every continuousf onK with|f| ≡1 can be uniformly approximated by continuous characters, i.e., there is a sequence{n_j}such thatαⁿ^j→f (α)uniformly for α∈K.

Hence forλ∈Sthere exists{n_j}such thatαⁿ^j→λforα∈K(uniformly). Ifμis a probability supported inK, we obtain thatμ(nˆ j)→λ. Whenμis strictly aperiodic, Corollary3.2applies andμhas the strong sweeping out property onT.

By Theorem 5.2.2(a) in [24],Tcontains a Cantor setKwhich is a Kronecker set. HenceK supports a continuous probabilityμwith uncountable support (so strictly aperiodic), and by the aboveμhas the strong sweeping out

property.

By Theorem3.1, a.e. convergence ofμⁿ∗f for everyf∈L₂(T, m)implies lim sup

ˆ

μ(n)→1,n =0

|1− ˆμ(n)|

1− | ˆμ(n)| <∞ (21)

for the Fourier–Stieltjes coefficients ofμ. We will prove the converse in Theorem3.6below.

First, let us observe that the proof of Corollary3.2shows that, if there is a sequence{λ_k} ⊂S∩σ (P )with 1 = λ_k→1, then (11) holds, contradicting (21). Hence, if (21) holds, thenσ (P )∩S =Sand therefore this intersection is finite, withσ (P^j)∩S= {1}for somej ≥1, sinceP is a positive contraction [14], Proposition 1. We give below a simple proof of this last fact for the convolution operators treated in this paper.