wacam’04july 17, 2004 a. fridgrowth of arithmetical complexity1 possible growth of arithmetical...

A. Frid Growth of arithmetical complexity1

WACaM’04 July 17, 2004

Possible growth of arithmetical complexity

Anna FridSobolev Institute of Mathematics

Novosibirsk, Russia

[email protected]://www.math.nsc.ru/LBRT/k4/Frid/fridanna.htm



Arithmetical closure

is the arithmetical closure of since

......21 nwwww

}0,0,|...{)( )1( ndkwwwwA dnkdkk

)(wA

).())(( wAwAA

w



Van der Waerden theorem

)(wA was invented by S. V. Avgustinovich in 1999

but

)(wAna

Theorem (Van der Waerden, 1927):

always contains arbitrarily long powers of some symbol .a

What else may occur in ?)(wA



A simple question

Does 010101… always occur in the Thue-Morse word?

0110 1001 1001 0110 1001 0110…

Proof:

Too precise question: in fact, any binary word does.

Avgustinovich, Fon-Der-Flaass, 1999

)()(, wAbawAba



Subword and arithmetical complexity

Subword complexity Arithmetical complexity)(nfw )(naw

)()( nanf ww nw,Growing functions

periodic ult. constant

non-periodic 1nW is complexity is

number of factors of w of length n

number of words of length n in A(w)



Possible growth?

How can complexity grow?subword

Many examples, no characterization

Is the question trivial? Maybe it is always exponential?

How can complexity grow?arithmetical

NO



Paperfolding word

P=0?1? – a pattern w=T(P,P,…)=T(P)

1 0 10 1 0 10w 0 01 110 0

aw(n)=8n+4 for n > 13

A generalization: Toeplitz words

the paperfolding word



First results and classification

Exponentialar. compl.

Linearar. compl.

Fixed pointsof uniform morphisms

Paperfolding wordThue-Morse word

[Avgustinovich, Fon-Der-Flaass, Frid, 00 (03)]



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)

• …



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)



Symmetric D0L words

0 011 10

Thue-Morse morphism,ar. compl. of the fixed point is 2n

0 0101 121

3 303

2 232

A symmetric morphism:

ii )0()(

Its fixed point is 010 121 010 121 232 121 ….

of ar. compl. 42 . 2n-2

In general, on the q-letter alphabet aw(n)=q2kn-2, k|q.



p-adic complexity

is the nimber of words occurring in subsequences of differences of w

Our technique does not work

)(nap kp

Open problem. What is of the Thue-Morse word?

)(3 na

0110 1001 1001 0110 1001 0110…



? ? ? ? ? ??? ? ? ? ? ? ???

Regular Toeplitz words

P1=ab?cd? a (3-regular) pattern

b ? c d ? baa c d ? a b ?? ?1P

P2=ef? a (3-regular) pattern

b e c d f baa c d e a b f? ?12 PP

A (3-regular) Toeplitz word

,...),(?...lim 2112 PPTPPPnn



Linearity

Theorem. Let w be a uniformly recurrent infinite word. Then aw(n)=O(n)

iff up 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

Uniformly recurrent word: all factors occur infinite number of times with bounded gaps



Another example

P=0?1? 2-regular paperfolding pattern

w=Q·T(P,P,…)=T(P1,P2, P1,P2,…), where

0 3 22 3 2 01w

P1=2?0?3?2?1?3?

Q=23? 3-regular

00 3233 2

P2=3?0?2?3?1?2?



Lowest complexity?

A word w is Sturmian if its subword complexity is minimal for a non-periodic word: .1)( nnfw

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?



Relatives of period doubling word

Let a be a symbol, p be a prime integer.Define

and

2w 0100 0101 0100 0100 0100 0101 0100 0100…period doubling word

3w 001001000 001001000 001001001… etc.

Rp(a)=ap-1?

wp=T(Rp(0),Rp(1),…, Rp(0),Rp(1),…)



Minimal ar. complexity

12262)(22

pp

n

na

pp pw

2 3

p

and these limits are minimal for uniformly recurrent words

[Avgustinovich, Cassaigne, Frid, submitted]



Plot for ar. complexity of wp

length

ar. c

om

pl.



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



More classification


Symm. D0L words

Linearar. compl.(un. rec.

characterized)

min

ar. compl.]))([log( nnfO pu

Sturmian words,O(n3)



Words with aw(n)=O(nfu([logp(n)]))

Recall that for a symbol a and a prime p

Rp(a)=ap-1?.

For u=u0u1…un… let us define

Wp(u)=T(Rp(u0),Rp(u1),…, Rp(un),…).

)(3 uw 000000001 000000001 000000000…

u=0010…



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 ...)011(2w ...).100(2w



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);



Particular cases - 2

• If fu(n)=O(an), then aw(n)=O(n1+log a); so, on the binary alphabet we can reach aw(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 between n log n and polynomials.

p

3



Geometric definition of Sturmian words

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

)1,0( )1,0[c



Subsequence of difference 2

Factors of an arithmetical subsequence also can be represented as intersections of a line with the grid



Dual picture: gates and faces

[Berstel, Pocchiola, 93]



Counting faces

By Euler formula,

f=e-v+1=

23

)1()1()()(2

1

1

nnnddn

n

d

where is the Euler function)(n

We have )(2/)( 3nOfnaw



Computational results

It seems that for 1/3< <2/3,

),(2/)( npfnaw

),( npwhere is a simple function, ultimately periodic on .n

For the Fibonacci word

2

15

,...9,8,9,8,9,8,9,8,11,10,8,5,3,1,0,0),( np



The current state


Symm. D0L words

Linearar. compl.(un. rec.

characterized)

min

ar. compl.]))([log( nnfO pu

Sturmian words,O(n3)

linear subword

complexity



Other complexities

• d-complexity, Ivanyi, 1987

• pattern complexity, Restivo and Salemi, 2002

• maximal pattern complexity, Kamae and Zamboni, 2002

• modified complexity, Nakashima, Tamura, Yasutomi, 1999

Only complexities which are not less than the subword one:



Thank you

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