Possible growth of arithmetical complexity

1. Anna FridSobolev Institute of MathematicsNovosibirsk, Russiafrid@math.nsc.ruhttp://www.math.nsc.ru/LBRT/k4/Frid/fridanna.htm Possible growth of arithmetical complexity Growth of arithmetical complexity

2. Growth of arithmetical complexity Arithmetical closure is the arithmetical closure of since

3. Growth of arithmetical complexity was invented by S. V. Avgustinovich in 1999 but Theorem (Van der Waerden, 1927):always contains arbitrarily long powers of some symbol Van der Waerden theorem What else may occur in ?

4. Growth of arithmetical complexity Too precise question: in fact, any binary word does. Proof: Avgustinovich, Fon-Der-Flaass, 1999 A simple question Does 010101… always occur in the Thue-Morse word? 0110 1001 1001 0110 10010110…

5. Growth of arithmetical complexity Arithmetical complexity Growing functions periodic ult. constant non-periodic W is complexity is Subword and arithmetical complexity Subword complexity number of factors of w of length n number of words of length n in A(w)

6. Growth of arithmetical complexity complexity grow? How can subword complexity grow? How can arithmetical Is the question trivial? Maybe it is always exponential? Possible growth? Many examples, no characterization NO

7. Growth of arithmetical complexity 0 0 1 0 1 0 1 0 1 0 1 1 0 1 Paperfolding word w=T(P,P,…)=T(P) P=0?1? – a pattern 0 the paperfolding word aw(n)=8n+4 for n > 13 A generalization: Toeplitz words

8. Growth of arithmetical complexity First results and classification Linearar. compl. Exponentialar. compl. Fixed pointsof uniform morphisms Thue-Morse word Paperfolding word [Avgustinovich, Fon-Der-Flaass, Frid, 00 (03)]

9. Growth of arithmetical complexity Arguments for arithmetical complexity Mathematics involved: • Van der Waerden theorem • more number theory: Legendre symbol, Dirichlet theorem, computations modulo p… (for words of linear complexity) • linear algebra (for the Thue-Morse word etc.) • geometry (for Sturmian words) • …

10. Growth of arithmetical complexity Further results • ar. compl. of fixed points of symmetric morphisms[Frid03] • characterization of un. rec. words of linear ar. compl.[Frid03] • uniformly recurrent words of lowest complexity[Avgustinovich, Cassaigne, Frid, submitted] • a family with ar. compl. from a wider class(new) • on ar. compl. of Sturmian words(Cassaigne, Frid, preliminary results published)

11. Growth of arithmetical complexity 0 01 1 10 0010 A symmetric morphism: 1121 2232 3303 Its fixed point is 010121010121232121 …. of ar. compl. 42 . 2n-2 Symmetric D0L words Thue-Morse morphism,ar. compl. of the fixed point is 2n In general, on the q-letter alphabet aw(n)=q2kn-2, k|q.

12. Growth of arithmetical complexity Open problem. What is of the Thue-Morse word? p-adic complexity is the nimber of words occurring in subsequences of differences of w 0110 1001 1001 0110 1001 0110… Our technique does not work

13. Growth of arithmetical complexity a b e c d f a b ? c d e a b f ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? a b ? c d ? a b ? c d ? a b ? A (3-regular) Toeplitz word Regular Toeplitz words P1=ab?cd? a (3-regular) pattern P2=ef? a (3-regular) pattern

14. Growth of arithmetical complexity Linearity Uniformly recurrent word: all factors occur infinite number of times with bounded gaps Theorem. Let w be a uniformly recurrent infinite word. Thenaw(n)=O(n) iffup to the set of factors w=T(P1,P2,…), where • all patterns Pi are p-regular for some fixed prime p; • sequence {P1,P2,…} is ultimately periodic

15. Growth of arithmetical complexity Another example P=0?1? 2-regular paperfolding pattern Q=23? 3-regular w=Q·T(P,P,…)=T(P1,P2, P1,P2,…), where 2 0 2 3 2 1 3 2 0 3 3 2 0 0 3 P1=2?0?3?2?1?3? P2=3?0?2?3?1?2?

16. Growth of arithmetical complexity Lowest complexity? A word w is Sturmian if its subword complexity is minimal for a non-periodic word: Is arithmetical complexity of a Sturmian word also minimal? NO, it is not even linear (Sturmian words are not Toeplitz words) What words have lowest ar. complexity?

17. Growth of arithmetical complexity 001001000001001000 001001001… etc. Relatives of period doubling word Let a be a symbol, p be a prime integer. Define Rp(a)=ap-1? and wp=T(Rp(0),Rp(1),…, Rp(0),Rp(1),…) 0100 0101 0100 0100 0100 0101 0100 0100… period doubling word

18. Growth of arithmetical complexity 2 3 Minimal ar. complexity and these limits are minimal for uniformly recurrent words [Avgustinovich, Cassaigne, Frid, submitted]

19. Growth of arithmetical complexity Plot for ar. complexity of w p ar. compl. length

20. Growth of arithmetical complexity Not uniformly recurrent? All results on linearity are valid only for uniformly recurrent word. Open problem. Are there (essentially) not uniformly recurrent words of linear arithmetical complexity? something un. rec. word is not considered

21. Growth of arithmetical complexity Linearar. compl.(un. rec. characterized) ar. compl. min More classification Sturmian words,O(n3) Exponentialar. compl. Symm. D0L words

22. Growth of arithmetical complexity For u=u0u1…un…let us define Wp(u)=T(Rp(u0),Rp(u1),…, Rp(un),…). Words with aw(n)=O(nfu([logp(n)])) Recall that for a symbol a and a prime p Rp(a)=ap-1?. u=0010… 000000001000000001 000000000…

23. Growth of arithmetical complexity A theorem Theorem. For all u (on a finite alphabet) and each prime p>2, aw(u)(n)=O(nfu([logp(n)])). for p=2, the situation is more complicated since 01010101... may occur both in and

24. Growth of arithmetical complexity Particular cases • If u is periodic, then aw(n)=O(n)which agrees with the characterization above; • If fu(n)=O(n), then aw(n)=O(n log n)for example, when u is a Sturmian word, or the Thue-Morse word, or 0 1 00 11 0000 1111…; • If fu(n)=O(n log n), then aw(n)=O(n log n log log n); • If fu(n)=O(na), then aw(n)=O(n (log n)a);

25. Growth of arithmetical complexity Particular cases - 2 • If fu(n)=O(an), then aw(n)=O(n1+log a);so, on the binary alphabet we can reachaw(n)=O(n1+log 2); for larger alphabets, the degree grows. • If fu(n) grows intermediately between polynomials and exponentials, then aw(n) grows intermediately betweenn log nand polynomials. p 3

26. Growth of arithmetical complexity is irrational, is arbitrary All Sturmian words can be constructed so, the set of factors does not depend on c, the subword complexity is n+1 Geometric definition of Sturmian words

27. Growth of arithmetical complexity Subsequence of difference 2 Factors of an arithmetical subsequence also can be represented as intersections of a line with the grid

28. Growth of arithmetical complexity Dual picture: gates and faces [Berstel, Pocchiola, 93]

29. Growth of arithmetical complexity Counting faces By Euler formula, f=e-v+1= where is the Euler function We have

30. Growth of arithmetical complexity For the Fibonacci word Computational results It seems that for 1/3< <2/3, where is a simple function, ultimately periodic on .

31. Growth of arithmetical complexity Linearar. compl.(un. rec. characterized) ar. compl. min The current state linear subword complexity Sturmian words,O(n3) Exponentialar. compl. Symm. D0L words

32. Growth of arithmetical complexity Other complexities Only complexities which are not less than the subword one: • d-complexity, Ivanyi, 1987 • pattern complexity, Restivo and Salemi, 2002 • maximal pattern complexity, Kamae and Zamboni, 2002 • modified complexity, Nakashima, Tamura, Yasutomi, 1999

33. Thank you Growth of arithmetical complexity