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WACaM’04
July 17, 2004
Possible growth of
arithmetical complexity
Anna Frid
Sobolev Institute of Mathematics
Novosibirsk, Russia
[email protected]
http://www.math.nsc.ru/LBRT/k4/Frid/fridanna.htm
A. Frid
Growth of arithmetical complexity
1
WACaM’04
Arithmetical closure
July 17, 2004

w

w
w
...
w
...


1
2
n
A
(
w
)

{
w
w
...
w
|
k
,
d

0
,
n

0
}
k
k

d
k

(
n

1
)
d
A(w) is the arithmetical closure
of w since
A
(
A
(
w
))

A
(
w
).
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Growth of arithmetical complexity
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WACaM’04
Van der Waerden theorem
July 17, 2004
A(w) was invented by S. V. Avgustinovich in 1999
but
Theorem (Van der Waerden, 1927):
A(w)
always contains arbitrarily long
n
a
powers
of some symbol a.
What else may occur in A(w)?
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WACaM’04
A simple question
July 17, 2004
Does 010101… always occur in
the Thue-Morse word?
0110 1001 1001 0110 1001 0110…
Too precise question: in fact, any
binary word does.
,
b

A
(
w
)

a

b

A
(
w
)
Proof: a
Avgustinovich, Fon-Der-Flaass, 1999
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WACaM’04
July 17, 2004
Subword and arithmetical complexity
Subword complexity
fw(n)
number of factors
of w of length n
Arithmetical complexity
a
)
w(n
number of words of
length n in A(w)
Growing functions
f
(
n
)

a
(
n
)
w
,n
w
w
W is
A. Frid
periodic
non-periodic

complexity is
ult. constant
n1
Growth of arithmetical complexity
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WACaM’04
Possible growth?
July 17, 2004
How can subword complexity grow?
Many examples, no characterization
How can arithmetical complexity grow?
Is the question trivial?
Maybe it is always exponential?
NO
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Growth of arithmetical complexity
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WACaM’04
Paperfolding word
P=0?1? – a pattern
w
w=T(P,P,…)=T(P)
0 0 1 0 0 1 1 0 0 0 1 1 0 1 1
the paperfolding word
aw(n)=8n+4
for n > 13
A generalization: Toeplitz words
A. Frid
July 17, 2004
Growth of arithmetical complexity
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WACaM’04
July 17, 2004
First results and classification
Exponential
ar. compl.
Fixed points
of uniform
morphisms
Linear
ar. compl.
Thue-Morse word
Paperfolding word
[Avgustinovich, Fon-Der-Flaass, Frid, 00 (03)]
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WACaM’04
July 17, 2004
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)
• …
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WACaM’04
Further results
July 17, 2004
•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)
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Growth of arithmetical complexity
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WACaM’04
0  01
1  10
0  010
1  121
2  232
3  303
Symmetric D0L words
July 17, 2004
Thue-Morse morphism,
ar. compl. of the fixed point is 2n
A symmetric morphism:

(
i
)


(
0
)

i
Its fixed point is
010 121 010 121 232 121 ….
of ar. compl. 42 . 2n-2
In general, on the q-letter alphabet
A. Frid
aw(n)=q2kn-2, k|q.
Growth of arithmetical complexity
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WACaM’04
p-adic complexity
July 17, 2004
ap(n) is the nimber of words occurring in
subsequences of differences pk of w
Open problem. What is a3(n)of the Thue-Morse
word?
0110 1001 1001 0110 1001 0110…
Our technique does not work
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Growth of arithmetical complexity
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WACaM’04
Regular Toeplitz words
July 17, 2004



?

P

P

?
P

?
 ?a ?b ?e ?c ?d ?f ?a ?b ?? ?c ?d ?e ?a ?b ?f
21
1 
P1=ab?cd?
a (3-regular) pattern
P2=ef?
a (3-regular) pattern

lim
P

...

P

P

?

T
(
P
,
P
,...
n
2
1
1
2
n


A (3-regular) Toeplitz word
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Growth of arithmetical complexity
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WACaM’04
Linearity
July 17, 2004
Uniformly recurrent word: all factors occur infinite number
of times with bounded gaps
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
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WACaM’04
Another example
P=0?1?
2-regular
Q=23?
3-regular
July 17, 2004
paperfolding pattern
w=Q·T(P,P,…)=T(P1,P2, P1,P2,…), where
w 2 3 0 2 3 0 2 3 1 2 3 0 2 3 0
P1=2?0?3?2?1?3?
P2=3?0?2?3?1?2?
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WACaM’04
Lowest complexity?
July 17, 2004
A word w is Sturmian if its subword complexity is
minimal for a non-periodic word: f
(
n
)

n

1
.
w
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?
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WACaM’04
July 17, 2004
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),…)
w2  0100 0101 0100 0100 0100 0101 0100 0100…
period doubling word
w3  001001000 001001000 001001001…
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etc.
WACaM’04
Minimal ar. complexity
July 17, 2004
a
(
n
)

2
w
2
p

2
6
p

2
p


p
n 2
p

1
p

2
3
and these limits are minimal for
uniformly recurrent words
[Avgustinovich, Cassaigne, Frid, submitted]
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WACaM’04
July 17, 2004
Plot for ar. complexity of wp
ar. compl.
length
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WACaM’04
Not uniformly recurrent?
July 17, 2004
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
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Growth of arithmetical complexity
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WACaM’04
July 17, 2004
More classification
Sturmian words,
O(n3)
Exponential
ar. compl.
Symm.
D0L words
Linear
ar. compl.
(un. rec.
characterized)
min
ar. compl.
O
(
nf
([log
n
]))
u
p
A. Frid
Growth of arithmetical complexity
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WACaM’04
July 17, 2004
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),…).
u=0010…
w
(
u
)000000001 000000001 000000000…
3
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WACaM’04
A theorem
July 17, 2004
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
w
(
011
...)
and
2
A. Frid
w
(
100
...).
2
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WACaM’04
Particular cases
July 17, 2004
• 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);
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WACaM’04
Particular cases - 2
July 17, 2004
• If fu(n)=O(an), then aw(n)=O(n1+log pa);
so, on the binary alphabet we can reach
aw(n)=O(n1+log3 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.
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Growth of arithmetical complexity
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WACaM’04
July 17, 2004
Geometric definition of Sturmian words

(
0
,
1
) is irrational, c
[
0
,
1
)is arbitrary
All Sturmian words can be constructed so, the set of
factors does not depend on c, the subword complexity is
n+1
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WACaM’04
July 17, 2004
Subsequence of difference 2
Factors of an arithmetical subsequence also can be
represented as intersections of a line with the grid

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WACaM’04
July 17, 2004
Dual picture: gates and faces
[Berstel, Pocchiola, 93]
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WACaM’04
Counting faces
By Euler formula,
July 17, 2004

(
n

1
)
n
(
n

1
)
(
n

d
)
(
d
)


2

f=e-v+1= 2
3
d

1
n

1
where (n)
We have
A. Frid
is the Euler function
3
a
(
n
)

f
/
2

O
(
n
)
w
Growth of arithmetical complexity
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WACaM’04
Computational results
July 17, 2004
It seems that for 1/3< <2/3,

a
(
n
)

f
/
2

p
(
n
,
)
w
(n
,
)is a simple function,
where p
ultimately periodic on n .

For the Fibonacci word


51


2 

p
(
n
,
)

0
,
0
,
1
,
3
,
5
,
8
,
10
,
11
,
8
,
9
,
8
,
9
,
8
,
9
,
8
,
9
,.
A. Frid
Growth of arithmetical complexity
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WACaM’04
July 17, 2004
The current state
Sturmian words,
O(n3)
Exponential
ar. compl.
Symm.
D0L words
linear
subword
complexity
Linear
ar. compl.
(un. rec.
characterized)
min
ar. compl.
O
(
nf
([log
n
]))
u
p
A. Frid
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WACaM’04
Other complexities
July 17, 2004
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
A. Frid
Growth of arithmetical complexity
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WACaM’04
July 17, 2004
Thank you
A. Frid
Growth of arithmetical complexity
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