AUTOMATIC SEQUENCES GENERATED BY SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE

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1 AUTOMATIC SEQUENCES GENERATED BY SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE JEAN-MARC DESHOUILLERS, MICHAEL DRMOTA, AND CLEMENS MÜLLNER Abstract. We prove that automatic sequences generated by synchronizing automata satisfy the full Sarnak conjecture. This is of particular interest, since Berlinkov [2, 3] proved recently that almost all automata are synchronizing.. Introduction In 2009, Peter Sarnak stated the following conjecture [7]: Conjecture.. Let µ be the Möbius function. For any sequence ξ(n) observed by a deterministic ow (X, T ), it holds that () ξ(n)µ(n) o(n). There are some classes of functions for which the Sarnak conjecture has already been proved, as for example periodic sequences, quasiperiodic sequences [7], nilsequences [2] and horocycle ows [5]. The purpose of this paper is to add a further class of sequences ξ(n) that fulll the Sarnak conjecture, namely automatic sequences that are generated by synchronizing automata. Let us be more precise. Suppose that A is a complete deterministic nite automaton with output (DFAO) with input alphabet Σ {0,..., k }, transition function δ, state set Q with the initial state q 0 and output mapping τ : Q. We say a sequence (a n ) n N is automatic if there exists a DFAO A such that a n τ(δ(q 0, w n )) holds for all n 0, where w n is the representation of n in base k. As in the denition given in [, p. 52], we assume the input starts with the most signicant digit. A DFAO A is called synchronizing if there exists a synchronizing word w Σ whose action resets A, i.e. w leaves the automaton in one specic state, no matter at which state in Q it is applied: δ(q, w) δ(q 0, w) for all q Q. Note that the output of a synchronizing automaton for an input word only depends on the last occurence of the synchronizing word and the part thereafter. We x a synchronizing DFAO A and denote by a the corresponding automatic sequence (a n ) n N. We denote by T the shift-operator on the sequences in. We then dene X : {T n (a) : n N 0 }. Since the subword complexity of an automatic sequence is at most linear it follows that the topological entropy of the dynamical system (X, T ) is zero. Thus, any automatic sequences (as well as any other sequence observed by (X, T )) is observed by a deterministic ow. It is therefore natural to study the Sarnak conjecture for automatic sequences and their related dynamical system. Actually a Möbius randomness law has already been established for the ThueMorse sequence [6], [5], the RudinShapiro sequence [6] and sequences related to invertible automata [8, 0].

2 2 JEAN-MARC DESHOUILLERS, MICHAEL DRMOTA, AND CLEMENS MÜLLNER As already mentioned the main focus of this paper is to study automatic sequences that are generated by synchronizing automata. It is worth mentioning that Berlinkov [2, 3] establishes a result showing that almost all automata are synchronizing. In particular this means that we are studying more or less almost all automatic sequences and only exceptional cases are not covered. (By the way the ThueMorse sequence as well the RudinShapiro sequence are exceptional from this point of view.) Nevertheless the following general result can be proved by elementary methods. Theorem.2. Let h : N C be a bounded function fullling (2) h(an + b) o(n) for all positive integers a, b. Let, furthermore, (X, T ) be the dynamical system related to an automatic sequence (a n ) n N that is generated by a synchronizing automaton, where the input starts with the most significant digit of n. Then for all sequences ξ(n) : f(t n (x)) (with x X and f C(X, C)) we have ξ(n)h(n) o(n). This implies the full Sarnak conjecture for this dynamical system. In Section 3 we address the case where the sequence under consideration is produced by a synchronizing automaton, where the input starts with the least signicant digit. Finally in Section 4 we discuss Gelfond-like problems for sequences that are generated by synchronizing automata. 2. Proof of Theorem.2 We recall that we work with a xed synchronizing DFAO A and denote by a the corresponding automatic sequence (a n ) n N. We denote by T the shift-operator on the sequences in and dene X : {T n (a) : n N 0 }. In this case we can use the metric d(x, y) n0 2 n d n (x n, y n ) on X, where d n denotes the discrete metric on. First we present a simple lemma that reduces the Sarnak conjecture for the dynamical system (X, T ) (i.e. the full Sarnak conjecture for a) to a Möbius random law on the corresponding automatic sequence a (a n ) n N. Lemma 2.. Suppose that for every j and for every function g : j C we have g(a n+l, a n+l+,..., a n+l+j )h(n) o(n), uniformly for l N, where h : N C satises (2). Then we also have ξ(n)h(n) o(n). for all sequences of the form ξ(n) f(t n x), where x X and f C(X, C). Proof. Let f C(X, C) and ε > 0 be given and suppose that h : N C satises (2) and w.l.o.g. h(n). Furthermore we assume that is linearly ordered.

3 SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE 3 By continuity there exists δ > 0 such that f(x) f(y) < ε/2 if d(x, y) < δ. Next x j with 2 j δ. For every x (x 0, x,...) X set x (j) (x 0, x,..., x j, y j, y j+,...) such that (y j, y j+,...) is the lexicographically smallest sequence in with (x 0, x,..., x j, y j, y j+,...) X. Then d(x, x (j) ) < 2 j δ and consequently f(x) f(x (j) ) < ε/2. Hence f(t n x)h(n) f((t n x) (j) )h(n) ε 2 N. Since x (j) only depends on x 0,..., x j the function f(x (j) ) depends only on x 0,..., x j, too, and can be written as f(x (j) ) g(x 0,..., x j ) for a properly chosen function g : j C. By assumption there exist N 0 N 0 (ε) such that g(a n+l, a n+l+,..., a n+l+j )h(n) ε 2 N for all N N 0 and all l 0. Finally we x x X. By denition this means that for all N N there exists l N with (3) In particular it follows that x n a n+l for n,..., N + j. f((t n x) (j) ) g(a n+l, a n+l+,..., a n+l+j ) for n,..., N, and, thus, f((t n x) (j) )h(n) g(a n+l, a n+l+,..., a n+l+j )h(n) ε 2 N. Consequently we have for N N 0 (ε) f(t n x)h(n) εn, which completes the proof of the lemma. Next we nd specic bounds on how often synchronizing words occur. Lemma 2.2. Let A be a synchronizing DFAO with synchronizing word w Σ m 0. There exists η > 0 depending only on m 0 and k such that the number of synchronizing words of length n is bounded from below by k n ( k m 0 ηn ) for all n N, that is, at most O(k n( η) ) words of length n are not synchronizing. The same kind of statement holds if we delete leading zeros of the words in Σ n and consider only those words in Σ n for which the reduced word is synchronizing. Proof. We note that if w occurs as a (consecutive) subword of some w Σ, then w is also a synchronizing word. We decompose now every word w Σ n into n/m 0 distinct parts, each of length m 0 and one part of length (n mod m 0 ). If w is not synchronizing, it follows directly that each of the blocks of length m 0 must not coincide with w. Thus we can bound the number of words of length n that are not synchronizing by (k m 0 ) n/m 0 k n mod m 0. We dene η : log(km 0 /(k m 0 )) > 0 and the statement follows directly. log(k0 m)

4 4 JEAN-MARC DESHOUILLERS, MICHAEL DRMOTA, AND CLEMENS MÜLLNER It is easy to extend the proof to the reduced case, where we delete leading zeros. Now we are prepared to prove the following proposition which gives rise to the main theorem of this paper, Theorem.2. Proposition 2.3. Let (a n ) n N be an automatic sequence generated by a synchronized automaton. Then for all j N, h : N C fullling (2) and g : j C g(a n+l, a n+l+,..., a n+l+j )h(n) o(n) uniformly for l N. Proof. Without loss of generality, we assume that g is absolutely bounded by. We have to show that ε > 0, N 0 N such that for all N N 0 and all l 0 g(a n+l, a n+l+,..., a n+l+j )h(n) εn, in particular, N 0 does not depend on l. We choose n such that there exist at least k n ( ε/(3k j )) synchronizing words of length n. (Lemma 2.2 shows that n ( ) m 0 log(ε/(3kj )) η log k is a valid choice.) Let M denote the set of integers 0 m < k n such that the j input sequences w m+i mod k n, 0 i < j, are synchronizing. Clearly we have M k n ( ε/3). We then choose N 0 such that for all a < k n, N N 0 h(n) ε N 3 k. n n a mod k n Furthermore we assume that N 0 3k n /ε. Note that N 0 only depends on n, k and ε(and h). Furthermore we note that if n m mod k n for some m < k n for which the input word w m is synchronizing we have a n a m since w n vw m for some v {0,..., k }. Hence, we obtain g(a n+l, a n+l+,..., a n+l+j )h(n) g(a n+l, a n+l+,..., a n+l+j )h(n) + #{n N : (n + l mod k n ) / M} m M n+l m mod k n N g(a m,..., a m+j ) h(n) + (k n M ) k m M n ε 3 N + ( N k n + ) k n ε 3 n m l mod k n εn,

5 SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE 5 which completes the proof. Of course this implies Theorem Reading the digits in reverse order We made the usual convention that we start with the most signicant digit of w n for the input in A. We could also start with the least signicant digit but then we will get (in general) another automatic sequence which can be also considered as an automatic sequence generated by a synchronizing automaton. If we start with the most signicant digit, then the analysis of Section 2 shows that the density of the pull-back of an element of the output set exists (see also Section 4). However, if we start with the least signicant digit the situation is completely dierent. As we will see in the analysis below we will typically get no density (just a logarithmic density). Nevertheless we can also say something in the direction of the Sarnak conjecture for sequences of this kind (however, in a slightly dierent form). Theorem 3.. Let h : N C be a bounded function fullling (4) h(n) o(n). Let, furthermore, (a n ) n N be an automatic sequence that is generated by a synchronizing automaton, where the input starts with the least signicant digit of n; without loss of generality we assume that C. Then we have a n h(n) o(n). This implies that (a n ) n N is orthogonal to the Möbius function in the sense (). In order to prove Theorem 3. we will use the following lemma. Lemma 3.2. Let k 2, f : N C a bounded function and B a subset of N with density. Then the following properties are equivalent: (5) f(n) o(n) (N ). (6) For all b B 0 n<k t f(bk t + n) o(k t ) (t ). Proof. Without loss of generality, we may assume that f is bounded by. First we assume (5) and prove (6) for any non-negative integer b not necessarily belonging to B. For ε > 0, we select T ε such that for all N k Tε ε 3b N. m N

6 6 JEAN-MARC DESHOUILLERS, MICHAEL DRMOTA, AND CLEMENS MÜLLNER For t T ε, we have f(bk t + n) n k t m (b+)k + t m b k t ε ( (b + )k t + bk t) 3b εk t, which proves (6). Second we now assume that (6) holds true. We x ε > 0 and construct an integer V such that (7) for all N k V f(n) εn. For this purpose we rst select v such that (8) for all M k v #([, M) B) ( ε ) M and k v ε 4 4. Condition (6) says that for all b B and for all η > 0 there exists T (b, η) such that for all t T (b, η) (9) f(bk t + n) 0 n<k ηkt. t We dene V by ( (0) V v + max T ε ) b,. b B, b k v+ 4k 2v+ Let N k V ; we let u log N/ log k so that k u N < k u+. We set A N/k u v so that k v A < k v+. We split the interval [, N] into the disjoint union of A + intervals I 0, I,..., I A given by I 0 [, k u v ), I a [ak u v, (a + )k u v ), for a < A, I A [Ak u v, N]. Since the length of any interval I i is at most k u v, we have () f(n) I 0 + I A 2k u v 2Nk v ε 2 N. n I 0 I A Since A k v, we have by (8) (2) f(n) n I a a<a, a B a<a, a B I a ε 4 Aku v ε 4 N.

7 SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE 7 If a B, we have by (9) and (0) f(ak u v + n) m I a 0 n<k u v Since A k v+, we have (3) a<a, a B m I a a<a, a B ε 4k 2v+ ku v m I a ε N. 4kv+ Aε 4k v+ N ε 4 N. Putting together (), (2), and (3), we obtain (7), which completes the proof of Lemma 3.2 It is now easy to complete the proof of Theorem 3.. Proof. (of Theorem 3.) First we observe that (4) and the rst part of Lemma 3.2 imply 0 n<k t h(bk t + n) o(k t ) for all positive integers b. Next let B be the set of positive integers b for which the k-ary expansions w b are synchronizing (when we start with the least signicant digit). Hence for every integer of the form m bk t + n (with b B and 0 n < k t ) the output a m of the automaton is constant, namely a b. Thus it follows that a bk t +nh(bk t + n) a b h(bk t + n) o(k t ) (t ). 0 n<k t 0 n<k t By Lemma 2.2 the set B has density. Consequently it follows by Lemma 3.2 that a n h(n) o(n), (N ) which nishes the proof. We remark that we (usually) get no densities of the pull-back of the elements of the output set. Namely, for every synchronizing b the set t 0 {bkt + n : 0 n < b t } (on which the output sequence equals a b ) has no density. Finally we note that we do not see how we could generalize Theorem 3. to the dynamical system related to (a n ) n N. We can show that for any j, g : j C and any l, one has g(a n+l, a n+l+,..., a n+l+j )h(n) o(n), (N ), but it is not clear how to show the uniformity in l. 4. Gelfond Problems Let s k (n) denote the k-ary sum-of-digits function. Then it is well known that sequences of the form a n s k (n) mod m are k-automatic. The most prominent sequence of this form is the ThueMorse sequence t n s 2 (n) mod 2.

8 8 JEAN-MARC DESHOUILLERS, MICHAEL DRMOTA, AND CLEMENS MÜLLNER Without using (or even knowing) the notion of automatic sequences, Gelfond [] proved that for every l {0,,..., m } #{n N : s k (an + b) l mod m} N m + O ( N η) for some η > 0 (provided that (k, m) ), that is, linear subsequences of the k-automatic sequence a n s k (n) mod m are asymptotically uniformly distributed on the output alphabet {0,,..., m }. What is even more interesting, in the same paper [] Gelfond formulated three problems, which are usually called Gelfond Problems: () If k, k 2 2 are coprime integers and (k, m ) (k 2, m 2 ) then #{n N : s k (n) l mod m, s k2 (n) l 2 mod m 2 } for some η > 0. (2) If k 2 with (k, m) then #{p N : p P, s k (p) l mod m} π(n) m N m m 2 + O ( N η) + O ( N η) for some η > 0. (Here π(x) denotes the number of primes x.) (3) If k 2 with (k, m) then for every integer polynomial P (x) for some η > 0. #{n N : s k (P (n)) l mod m} N m + O ( N η) Whereas the rst problem was almost immediately solved by Besineau [4] (without an explicit error term; the error term was nally proved by Kim [3]) it took more than 40 years till the other two problems were solved or came close to a solution. Actually, the second problem on the subsequence along the primes was solved by Mauduit and Rivat [5]. The third problem was completely solved for quadratic polynomials by Mauduit and Rivat [4] and partially solved for general polynomials by Drmota, Mauduit and Rivat [9] (it is assumed that the base k is prime and suciently large with respect to the degree of P (x)). It is immediate to translate the Gelfond problems to arbitrary automatic sequences. The only dierence is that we cannot expect a uniform distribution on the output alphabet but just densities (or only logarithmic densities). The general question is maybe out of reach, nevertheless we can give a complete answer for automatic sequences that are generated by synchronizing automata. Theorem 4.. Suppose that k > and k 2 > are coprime integers and (a n ) n N N and (b n ) n N N 2 are automatic sequences that are generated by a synchronizing k -automaton and a synchronizing k 2 -automaton, respectively. Then for every pair (α, β) (, 2 ) the density δ(α, β) lim N N #{n N : (a n, b n ) (α, β)} exists and equals the product of the densities of α in (a n ) n N and β in (b n ) n N.

9 SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE 9 Theorem 4.2. Suppose that (a n ) n N N is an automatic sequence that is generated by a synchronizing automaton. Then for every α the density δ P (α) lim N π(n) #{p P, p N : a p α} exists. Theorem 4.3. Suppose that (a n ) n N N is an automatic sequence that is generated by a synchronizing automaton and P (x) is a non-zero and positive integer valued polynomial. Then for every α the density δ P (α) lim N N #{n N : a P (n) α} exists. Before we start with the proof of Theorems we show that every element of the output alphabet of an automatic sequence that is generated by a synchronizing automaton has a density. Lemma 4.4. Suppose that (a n ) n N N is an automatic sequences that is generated by a synchronizing automaton. Then for every α the density δ(α) lim N N #{n N : a n α} exists. Proof. Let ε > 0 be given. Then by using a simplied version of the arguments in the proof of Proposition 2.3 there exists n and a set M of integers 0 m < k n such that M k n ( ε) and that all words w m with m M are synchronizing (in the automaton related to (a n ) n N ). We also recall that a n a m if n m mod k n. Hence it follows that #{n N : a n α} #{n N : n m mod k n } + O(εN) ( N k n ) + O() + O(εN) #{m M : a m α} N + O(k n ) + O(εN). k n Consequently, the sequence N #{n N : a n α} is a Cauchy sequence and, thus, convergent. Proof. (of Theorem 4.) As in the proof of Lemma 4.4 we construct for given ε > 0 a set M of integers 0 m < k n such that M k n ( ε) and that all k -ary words w m with m M are synchronizing in the automaton related to (a n ) n N. In the same way we can handle the sequence (b n ) n N. Therefore we have to construct a corresponding set M 2 of integers 0 m 2 < k n 2 2 such that M 2 k n 2 2 ( ε) and that all k 2 -ary words w m2 with m 2 M 2 are synchronizing in the automaton related to (b n ) n N.

10 0 JEAN-MARC DESHOUILLERS, MICHAEL DRMOTA, AND CLEMENS MÜLLNER Similarly to the above we obtain #{n N : (a n, b n ) (α, β)} #{n N : n m mod k n, n m 2 mod k n 2 2 } + O(εN) m M, a m α m 2 M 2, b m2 β m M, a m α m 2 M 2, b m2 β #{m M : a m α} k n ( N k n k n 2 2 ) + O() + O(εN) #{m 2 M 2 : b m2 β} k n N + O(k n 2 k n 2 2 ) + O(εN), 2 where we have used that k and k 2 are coprime. Hence the existence of δ(α, β) follows, and also that it is equal to δ(α) δ(β). Proof. (of Theorem 4.2) For every ε > 0 we dene a set M as in the proof of Lemma 4.4. Furthermore we denote by π(n) the number of primes p N and by π(a, k; N) the number of primes p N with p a mod k. By the prime number Theorem in arithmetic progressions we have π(a, k; N) π(n)/ϕ(k) if (a, k). Here we have #{p P, p N : a p α} #{p P, p N : p m mod k n } + O(επ(N)) m M, (m,k n ), a mα which shows that the sequence π(n) #{p P, p N : a p α} is a Cauchy sequence and, thus, convergent. π(n) ϕ(k n ) + o(π(n)) + O(k n ) + O(επ(N)), Proof. (of Theorem 4.3) The proof is (again) very similar to the proof of Lemma 4.4. With the help of the notation we have #{n N : a P (n) α} c(m, K) #{0 n < K : P (n) m mod K} #{n N : P (n) m mod k n } + O(εN) ( c(m, k n ) N ) k + O(c(m, n kn )) + O(εN) c(m, k n ) k n N + O(k 2n ) + O(εN). Again this proves convergence and completes the proof of the theorem. Acknowledgements. All authors are supported by the binational research project MuDeRa which is funded by the French and Austrian Science Funds ANR and the FWF. The last two authors are also supported by the project S55-02 of the Austrian Science Fund FWF

11 SYNCHRONIZING AUTOMATA FULFILL THE SARNAK CONJECTURE which is part of the Special Research Program Quasi-Monte Carlo Methods: Theory and Applicaitons. References [] J.-P. Allouche and J. Shallit. Automatic sequences. Theory, applications, generalizations. Cambridge: Cambridge University Press, [2] M. V. Berlinkov. On the probability to be synchronizable. preprint, arxiv: , 203. [3] M. V. Berlinkov. Testing for synchronization. preprint, arxiv: , 204. [4] J. Bésineau. Indépendance statistique d'ensembles liés à la fonction somme des chires. Acta Arith., 20:4046, 972. [5] J. Bourgain, P. Sarnak, and T. Ziegler. Disjointness of Moebius from horocycle ows. In From Fourier analysis and number theory to Radon transforms and geometry, volume 28 of Dev. Math., pages Springer, New York, 203. [6] C. Dartyge and G. Tenenbaum. Sommes des chires de multiples d'entiers. Ann. Inst. Fourier (Grenoble), 55(7): , [7] H. Davenport. On some innite series involving arithmetical functions (II). The Quarterly Journal of Mathematics, os-8():33320, 937. [8] M. Drmota. Subsequences of automatic sequences and uniform distribution. In Uniform distribution and quasi-monte Carlo methods, volume 5 of Radon Ser. Comput. Appl. Math., pages De Gruyter, Berlin, 204. [9] M. Drmota, C. Mauduit, and J. Rivat. The sum-of-digits function of polynomial sequences. J. Lond. Math. Soc., 84():802, 20. [0] S. Ferenczi, J. Kuªaga-Przymus, M. Lemanczyk, and C. Mauduit. Substitutions and Möbius disjointness. preprint, arxiv: v, 205. [] A. Gelfond. Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith., 3:259265, 967/968. [2] B. Green and T. Tao. The mobius function is strongly orthogonal to nilsequences. Ann. of Math. (2), 75(2):54566, 202. [3] D.-H. Kim. On the joint distribution of q-additive functions in residue classes. J. Number Theory, 74(2):307336, 999. [4] C. Mauduit and J. Rivat. La somme des chires des carrés. Acta Math., 203():0748, [5] C. Mauduit and J. Rivat. Sur un problème de Gelfond: la somme des chires des nombres premiers. Ann. of Math. (2), 7(3):59646, 200. [6] C. Mauduit and J. Rivat. Prime numbers along Rudin-Shapiro sequences. preprint. [7] P. Sarnak. Three lectures on the Mobius function randomness and dynamics. address: jean-marc.deshouillers@math.u-bordeaux.fr J.M.D., Institut Mathématique de Bordeaux, Bordeaux INP, Talence, France address: michael.drmota@tuwien.ac.at Institut fur Diskrete Mathematik und Geometrie TU Wien, Wiedner Hauptstr. 80, 040 Wien, Austria address: clemens.muellner@tuwien.ac.at Institut fur Diskrete Mathematik und Geometrie TU Wien, Wiedner Hauptstr. 80, 040 Wien, Austria