Irregular $\mathcal{B}$-free Toeplitz sequences via Besicovitch's construction of sets of multiples without density

Modifying Besicovitch's construction of a set $\mathcal{B}$ of positive integers whose set of multiples $\mathcal{M}_{\mathcal{B}}$ has no asymptotic density, we provide examples of such sets $\mathcal{B}$ for which $\eta:=1_{\mathbb{Z}\setminus\mathcal{M}_{\mathcal{B}}}\in\{0,1\}^{\mathbb{Z}}$ is a Toeplitz sequence. Moreover our construction produces examples, for which $\eta$ is not only quasi-generic for the Mirsky measure (which has discrete dynamical spectrum), but also for some measure of positive entropy. On the other hand, modifying slightly an example from Kasjan, Keller, and Lema\'nczyk, we construct a set $\mathcal{B}$ for which $\eta$ is an irregular Toeplitz sequence but for which the orbit closure of $\eta$ in $\{0,1\}^{\mathbb{Z}}$ is uniquely ergodic.


Introduction
For subsets B Ď AEzt1u denote by M B :" Ť bPB b the set of all multiples of B. The density dpM B q :" lim NÑ8 N ´1 cardpB X r1 : Nsq exists in many cases, in particular if B is thin, i.e. if ř bPB 1{b ă 8, but Besicovitch [1] provided examples where it does not exist.A bit later, Davenport and Erdős [3] proved that the logarithmic density of M B always exists and coincides with the lower density dpM B q.For more background on this see [6,Sec. 2.5].
Recurrence properties of the sequence η " 1 F B :" 1 zM B P t0, 1u can be studied using dynamical systems theory, see in particular [6].To that end denote by S the left shift on t0, 1u and restrict this homeomorphism to the closure X η of tS n η : n P u in t0, 1u .There is a distinguished invariant measure µ η on X η (its Mirsky measure) for which η is quasi-generic.Indeed, N ´1 ř N n"1 δ S n η converges weakly to µ η along each subsequence pN i q i along which the lower density dpM B q is attained.Hence η is generic for µ η if and only if dpM B q exists.It follows that for the sets B constructed by Besicovitch [1], the point η is quasi-generic for at least one further invariant measure on X η .
In [6] and [9] properties of the topological and measure theoretic dynamical systems pX η , S q and pX η , S , µ η q were characterized in (elementary) number theoretic terms.Combining some of these results, it turns out that Besicovitch's examples always lead to proximal systems pX η , S q, namely to systems where each closed invariant subset contains the fixed point 0 , and that such systems have positive topological entropy and host a huge collection of ergodic invariant measures, among them a unique measure of maximal entropy, see Remark 2 in Section 4. But there are also plenty of sets B for which the system pX η , S q itself is minimal.In these cases, η is a Toeplitz sequence (see Remark 1), and it is hitherto unknown whether there are examples of this type where X η can host more invariant measures than just the Mirsky measure. 1ollowing Hall [8], we call B a Besicovitch set, if the density dpM B q exists.In this note we modify Besicovitch's construction and prove the the following result: 2Theorem 1.There are sets B Ď AEzt1u with the following properties: i) The sequence η " 1 F B is an irregular Toeplitz sequence.
ii) The set of shift invariant measures on X η contains at least one measure of positive entropy.iii) Depending on the details of the construction one can make sure that (a) B is not a Besicovitch set and η is quasi-generic for some measure of positive entropy, or (b) B is a Besicovitch set, so η is generic for the Mirsky measure, but there is also some measure of positive entropy.If ω is a regular Toeplitz sequence, then pX ω , S q is uniquely ergodic and has entropy zero [4, Thm.2.5].For irregular Toeplitz sequences ω a wide range of different dynamical properties of pX ω , S q is possible, see e.g.[15], [2], [4,Ex. 5.1,6.1].For irregular B-free Toeplitz sequences η at least one of these dynamical possibilities is excluded, namely to have a uniquely ergodic system pX η , S q of positive entropy (cf.[2]), because the Mirsky measure always exists and has entropy zero.
Other examples of irregular B-free Toeplitz sequences were provided in [9,Ex. 4.2].It is not immediately clear from that construction, however, whether those examples may/must possess at least two invariant measures, or whether they even may/must have positive entropy.Here we modify and tune that construction in such a way that we end up with an irregular Toeplitz sequence η for which pX η , S q is uniquely ergodic.I am indebted to Stanisław Kasjan, who provided a more systematic description of the construction from [9, Ex. 4.2], which was instrumental in proving the following theorem.
Theorem 2. There are sets B Ď AEzt1u with the following properties: i) The sequence η " 1 F B is an irregular Toeplitz sequence.ii) X η is uniquely ergodic, in particular of entropy zero.
Before we turn to the proof of Theorem 1 in Section 5, we prove Theorem 2 in Section 2, provide a useful "prime number characterization" of those sets B for which η is a Toeplitz sequence in Section 3, and prove a simplified version of Theorem 1 (existence of at least two invariant measures for which η is quasi-generic) in Proposition 2 of Section 4. The proof of one lemma, for which we rely on properties of Kolmogorov complexity, is deferred to Section 6.

Proof of Theorem 2
The starting point of the construction is a sequence pP k q kPAE of finite sets of prime numbers satisfying (I) P 1 " t2u, 3 , and (III) dpM P k q ě 1 ´2´pk`2q for k ě 2.
It is convenient to denote the elements of P k by q pkq 1 , . . ., q pkq t k , so cardpP k q " t k and Q k " ś t k s"1 q pkq s .Observe that t 1 " 1 and q p1q 1 " 2. Once the numbers t 1 , t 2 , . . .are fixed we can choose the second basic ingredient of the construction, namely we fix 4(IV) for each k P AE, a partition of AE into pairwise disjoint infinite sets R pi 1 ,...,i k q where i ℓ P t1, . . ., t ℓ u for all ℓ " 1, . . ., k, and such that R p1q " R pt 1 q " AE and After these preliminaries we define positive integers for j, k P AE such that k `j P R pi 1 ,...,i k q , As Ť pi 1 ,...,i k q R pi 1 ,...,i k q " in view of (IV), this defines the numbers c pkq k`j for all j, k P AE.Finally let b 1 " pq p1q 1 q 3 " 2 3 , and, for It follows that Lemma 1.The sequence η " 1 F B is an irregular Toeplitz sequence and B is thin, i.e. ř bPB 1{b ă 8.
Proof.Let S k " tb 1 , . . ., b k u and define A S k :" tgcdpℓ k , bq : b P Bu.Observe that in particular lim sup kÑ8 pA S k zS k q " H, so that η is a Toeplitz sequence by [9, Thm.B].As each set R pi 1 ,...,i k q is infinite, we have indeed In order to prove that η is irregular it suffices to show that inf see [9,Lem. 4.3] together with [4,Thm. 2.5].Observe first that dpM B q ď . This shows in particular that B is thin, so that the density dpM B q exists.
Next, observing (III) and the fact that the P k are pairwise disjoint sets of prime numbers, Hence the term in ( 3) is lower bounded by 1 2 ´1 16 ´1 4 " 3 16 .this shows that η is irregular.
and, as S k has period at most ℓ k and as where the last estimate is based on the following observation: If mb k`r , nb k`s P I " ra, a `Lq for some 2 ď r ă s and m, n P , then 0 ă |mb k`r ´nb k`s | ă L so that gcdpb k`r , b k`s q ă L. However, where k `s P R pi 1 ,...,i k`r q , so that, also in view of (II) and ( 1), In particular, µtx P X η : x 0 " 1u ě 1´dpM B q for each invariant measure µ on X η .But in view of [11,Thm. 4] (which owes much to Moody [13]) and the correspondence between the "sets of multiples" and "the cut and project" points of view on B-free numbers (see [9], in particular Lemma 4.1), the Mirsky measure is the only invariant measure on X η which satisfies this inequality.Hence pX η , S q is uniquely ergodic.

Non-uniquely ergodic B-free Toeplitz sequences
For each ε ą 0, Besicovitch [1] provided an example of a primitive set G Ď AEzt1u such that the lower asymptotic density dpM G q ă ε, while the upper asymptotic density of this set is dpM G q ą 1 2 . 6emark 2. The set M G in Besicovitch's example contains arbitrarily long intervals rT, 2T q.Since, by the Bertrand postulate (proved by Tchebichef [14, pp. 371-382]) each such interval contains at least one prime number, the set G contains infinitely many prime numbers, and so the corresponding subshift Our goal is to modify Besicovitch's construction in several respects by defining primitive sets -there is at least one invariant measure of positive entropy for which η is not quasi-generic, and -depending on details of the construction, η is generic for the Mirsky measure, or it is quasi-generic for some measure of positive entropy (and, of course, for the Mirsky measure).
We start by recalling the essentials of Besicovitch's construction, following more or less the outline in [8, second part of Thm.0.1]: Take positive numbers ε, ε i pi " 1, 2, . . .q such that ε ă 1 4 , Denote E T :" M rT,2T q and write epT q for the asymptotic density of E T .As E T is periodic, there are numbers λpT q such that the mean density of the set E T on any interval of more than λpT q consecutive integers is ă 2epT q.
Define integers 1 " T 0 ă T 1 ă T 2 ă T 3 ă . . .so that T 3 ą λpT 2 q, epT 3 q ă ε 3 . . .These inductive choices are possible, because of Erdős' result [7] that lim T Ñ8 epT q " 0. 7 Observe that, given T 1 , . . ., T k , the index T k`1 can be chosen arbitrarily large.We will make use of this freedom of choice in the sequel. 8 Besicovitch's set G is then defined as and obviously rT k , 2T k q Ď Ť 8 j"1 E T j " M G for all k.As Besicovitch observed, We now proceed to introduce additional constraints to the choice of the indices T k and to construct a set B Ď AEzt1u with the following properties: (I) For every j P AEzB there is a finite set P j of primes such that B{ j Ď M P j (see also Section 3), (II) dpM B q ă ε, and (III) dpM B q ě 1 2 ´2ε.To this end assume that integers 1 " T 0 ă T 1 ă ¨¨¨ă T k , positive integers L 1 , . . ., L k , and finite sets P 1 , . . ., P T k ´1 of prime numbers are chosen such that (setting T ´1 " 1) 7 Besicovitch [1] used his weaker Theorem 1, which asserts that ep2 1 q `ep2 2 q `¨¨¨`ep2 n q " opnq. 8Instead of the constraint T k ą λpT k´1 q in (4), Besicovitch requires more explicitly 2 i k ą p2 i k´1 `1q!.
(A) the following strengthening of Besicovitch's constraints (4) is satisfied for i " 1, . . ., k: T i ě L i ą λpT i´1 q, epT i q ă ε i , (B) cardp j ¨FP j X rT, 2T qq ď 2dp j ¨FP j q ¨T for all j P r1, T k´1 q and T ě T k , (C) Specpr1, 2T k qq Ď P j for all j P rT k´1 , T k q, and (D) dp j ¨FP j q ă ε ¨2´p j`1q for all j P r1, T k q.
Observe first that conditions (A) -(D) are empty and hence trivially satisfied for k " 0. Now we choose T k`1 ě L k`1 ą maxtT k , λpT k qu and sets P j pT k ď j ă T k`1 q inductively in such a way that (A) -(D) hold for k `1 instead of k: First we make sure that T k`1 is large enough to satisfy (A) and (B) for k `1.(For property (B) note that the sets j ¨FP j are periodic.)Then we choose the additional P j big enough such that also (C) and (D) are satisfied for k `1.
For the next step of the construction we fix, for all k P AE, sets J k Ď rT k , T k `Lk q (with additional properties to be specified below), and define  (6).
We will use the following two estimates: Lemma 6.For all k P AE, Here the first "0-sum" is due to property (C), and for the second "0-sum" one only needs to observe that 1 R F P j .The final estimate uses property (D).b) Proposition 2. There are primitive sets B with the following properties: i) The sequence η " 1 F B is a Toeplitz sequence.ii) B is not a Besicovitch set.
iii) The sequence η is quasi-generic for at least two measures.
Proof.Let J k " rT k , 2T k q for all k.Lemma 5d) shows that η is a Toeplitz sequence, and Lemma 5c) and Lemma 6a) imply for every k, so that in particular dpM B q ą 1 2 ´ε.Combined with Lemma 5d) this shows that B is not a Besicovitch set and that η is not generic for any measure, so it is quasi-generic for at least two measures.

Positive entropy
For the proof of Proposition 2 we made the straightforward choice J k " rT k , 2T k q.In order to control the entropy of the measures we construct, we will have to make more subtle choices for the sets J k Ď rT k , 2T k q, and in order to include also measures, for which η is not quasi-generic, we replace the intervals rT k , 2T k q by more flexible intervals rT k , T k `Lk q.The choice of the sets J k is based on the following lemma, which might be folklore among specialists, but which I could not locate in the literature.So I provide a proof based on properties of Kolmogorov complexity in Section 6. Lemma 7. Let ε P p0, 1  2 q.There is a constant L ε ą 0 such that for all L ě L ε and γ P p0, 1{2 ´εq there is a word w L,γ P t0, 1u L with the following properties: For each n ą 0 and each κ ą 0 there is ℓ n,κ ą 0 such that, for all sets A, B Ď t1, . . ., Lu with d A , d B ă ε, Φpd A q, Φpd B q ă 1 4 κ and w L,γ ¨1A c ¨1B " 0, We now describe how to choose the J k in order to get a measure of positive entropy for which η is quasi-generic.So let ε P p0, 1  4 q and choose γ P pε, 1 2 ´εq.Fix also some number κ P p0, εq.For each n P AE and all indices k such that L k ě ℓ n,κ we choose a word w k " w L k ,γ P t0, 1u n as in Lemma 7.
For any w P t0, 1u L denote Jpwq :" ti P r1, Ls : w i " 1u.Define the sets J k for our construction, Fix a subsequence pT k i q i for which j"0 δ S j η ¯i converges weakly to some invariant measure ν 1 , and δ S j η ¯i converges weakly to some invariant measure ν 2 .Then, δ S j η ¯i converges weakly to the invariant measure ν " 1  2 pν 1 `ν2 q, and δ S j η ¯i converges weakly to the invariant measure ν 1 .Without loss of generality we may assume that pT k i q i is the full sequence pT k q k -this just eases the notation.
Lemma 8. We have the following lower bound for the Kolmogorov-Sinai entropy of pX η , S , ν 2 q: with sets A k , B k Ď r1, L k s such that B k X pJpw k qzA k q " H, to which we want to apply Lemma 7with 2ε instead of ε and κ " 4Φp2ǫq.To this end observe that Lemma 6 implies in particular also Φpd A k q, Φpd B k q ă Φp2εq " κ 4 .Hence, Lemma 7 shows that for each n P AE there is So fix n P AE.For each cylinder set rus determined by u P t0, 1u n we have It follows from ( 10) and ( 12) that H n pν 2 q, the entropy of ν 2 on blocks of length n, can be estimated by 1 n H n pν 2 q ě Φpγq ´κ " Φpγq ´4Φp2ǫq, so that h ν 2 pS q ě Φpγq ´4Φp2εq.
Proof of Theorem 1. (ii) By Lemma 8, h ν 2 pS q ě Φpγq´4Φp2ǫq is strictly positive if γ ą Φ ´1p4Φp2ǫqq, which can easily be achieved for small enough ε ą 0. (i) η is a Toeplitz sequence by Lemma 4d).It is irregular, because X η is not uniquely ergodic by assertion (ii).
(iii) (a) Choose T k " L k .Then η is quasi-generic for the invariant measure ν " 1 2 pν 1 `ν2 q, and h ν pS q ě 1 2 h ν 2 pS q ą 0 as above.(b) Choose T k " k 2 L k .Then ν 2 is a measure of positive entropy as before, but the set B is Besicovitch:

On Kolmogorov complexity and entropy
Only Lemma 7 from this section will be used in the sequel.It is formulated just in terms of entropy, and the reader who considers it as folklore should skip this section.Very loosely speaking, the Kolmogorov complexity Cpwq of a word w P t0, 1u ˚is the length of the shortest binary code that can serve as a program for a universal Turing machine to print the word w on its output tape and then to stop.Of course this definition depends on the choice of the particular Turing machine, but it can be shown that for any two different universal Turing machines there exists a constant such that the difference of complexities defined with respect to these two machines does not exceed this constant for any word w of any length.The monograph [12] provides a precise and detailed introduction to Kolmogorov complexity and other variants of algorithmic complexity and their relation to entropy and coding, and we will refer to notation and results from this book throughout this section.
A general pitfall when dealing with algorithmic complexity is that (in)equalities which one might expect when one does not think too much about the details of their proofs, hold only up to a constant or even logarithmic (logarithm of the word length) error term.One of the reasons is that the transitions between consecutive words on the same input tape of the Turing machine must be recognizable, another one that sometimes the word length must be provided as additional information to the Turing machine to make the intended algorithm work.This can be dealt with properly by introducing variants of Kolmogorov complexity like the prefix complexity Kpwq in [12,Sec. 3.1].It should not come as a surprise that Cpwq and Kpwq differ only by a logarithmic (in the word length) term.As logarithmic terms do not influence our arguments, we will provide only "naive" proofs, whenever complexity is involved.
Proof.For large enough L ("large" depending only on ε) we can fix k P AE such that γ `ε{2 ă k{L ă γ `ε.Hence there are at least `L k ˘words w P t0, 1u L with pγ `ε{2qL ď ř L i"1 w i ď pγ `εqL.At least one of these words has complexity Cpwq ě log 2 `L k ˘´1 [12, Thm.2.2.1], and one can estimate that this is bounded from below by LΦpγq when L ě L ε for some suitable L ε . 9e fix some notation.
(1) For s P t0, . . ., n ´1u denote by H s n pwq the entropy of the empirical distribution of blocks of length n in the sample pw r jn`s`1, jn`s`ns q j"0,...,m´1 .(These are the non-overlapping sub-words of w with length n starting at position s, except possibly for the last one.)(2) Denote by H n pwq the entropy of the empirical distribution of blocks of length n in the sample pw r j`1, j`ns q j"0,...,L´n .(These are all sub-words of w with length n.) -For A Ď t1, . . ., Lu denote d A :" cardpAq{L.
Proof.Denote by w 1 the restriction of the word w to the indices r1, L 1 s.Then the collection of length-n subword of w 1 is the disjoint union of the samples from item (1), so that H n pw 1 q ě 1 n ř n´1 s"0 H s n pwq, because the entropy function (as a function on probability vectors) is concave.So it remains to estimate the difference H n pw 1 q ´Hn pwq.As L ´L1 ă n, a crude estimate can use the fact that the relative frequencies of any block u P t0, 1u n in w and w 1 can differ by at most pn ´1q{L 1 ă n{ppm `1qnq " 1{pm `1q.Hence, the contribution of each single block to the entropy can change by at most ϕp1{pm `1qq, where we use that the function ϕpxq " ´x log 2 x is concave and increasing on the interval r0, e ´1s.It follows that |H n pw 1 q ´Hn pwq| ď 2 n ϕp1{pm `1qq ď 2 n ϕp n L´pn´1q q ": q n pLq and q n pLq Ñ 0 as L Ñ 8.
Sketch of a proof of Lemma 7 using Kolmogorov complexity.The inequalities in (7) are obvious.We turn to the lower bound for the entropy.Let w " w L,γ .It is intuitively clear that Cpwq ď Cpw ¨1A c `1B q `Cp1 B q `Cpw ¨1A q `Oplog Lq.
But H 1 p1 B q ď Φpd B q, H 1 pw ¨1A q ď Φpd A q, and 1 n ř n´1 s"0 H s n pw ¨1A c `1B q ď H n pw ¨1A c `1B q `qn pLq by Lemma 10, so that Cpwq ď L ˆ1 n H n pw ¨1A c `1B q `m L ǫ n pmq `qn pLq `Φpd B q `Φpd A q `ǫ1 pLq ˙.

Lemma 4 .
a) B is primitive by construction.b) B X F " H by construction.c) B{ j Ď M P j for every j P AEzB.d) η " 1 F B is a Toeplitz sequence.Proof.a) and b): Obvious.c) Let b P B{ j.Then jb P B, whence jb R F by assertion b).In particular, jb R j ¨F P j , i.e. b R F P j .Hence b " 1 or b P M P j .But b ‰ 1 since j R B. d) η is a Toeplitz sequence by Proposition 1 and assertions a) and c).

3
Another characterization of the case when η is a Toeplitz sequence A set B Ď AE is primitive, if no number from B divides another one.If B is not primitive, there is always a unique primitive subset B 1 Ď B such that M B " M B 1 .Lemma 3. Suppose that B is primitive and let k P AEzB.Then B{k contains no infinite pairwise coprime subset if and only if there is a finite set of primes P k such that B{k Ď M P k .Proof.B{k contains an infinite pairwise coprime subset if and only if B{k Ę M C for all finite setsC Ď AEzt1u [6, Thm.3.7]5, and the latter is equivalent to B{k Ę M P for all finite sets P of primes.The claim of the lemma is just the negation of this equivalence.Proposition 1. Suppose that B is primitive.The sequence η " 1 F B is a Toeplitz sequence if and only if for every k P AEzB there is a finite set P k of primes such that B{k Ď M P k .Proof.η is a Toeplitz sequence if and only if there are no k P AE and no infinite pairwise coprime set A Ď AEzt1u such that kA Ď B [9, Thm.B].As B is primitive, there can never be k P B and an infinite pairwise coprime set A Ď AEzt1u such that kA Ď B. Hence η is a Toeplitz sequence if and only if there are no k P AEzB and no infinite pairwise coprime set A Ď AEzt1u such that kA Ď B. But kA Ď B is equivalent to A Ď B{k, so that an application of Lemma 3 finishes the proof.
For k P AE let B{k :" t b k : b P B, k | bu.Observe that B{k " t1u if and only if k P B, whenever B is primitive.