Possible growth of arithmetical complexity

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Possible growth of arithmetical complexity

• Anna FridSobolev Institute of MathematicsNovosib
  irsk, Russiafrid_at_math.nsc.ruhttp//www.math.nsc
  .ru/LBRT/k4/Frid/fridanna.htm

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Arithmetical closure
is the arithmetical closure of since
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Van der Waerden theorem
What else may occur in ?
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A simple question
Does 010101 always occur in the Thue-Morse word?
0110 1001 1001 0110 1001 0110
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Subword and arithmetical complexity

• Subword complexity

• Arithmetical complexity

number of factors of w of length n
number of words of length n in A(w)
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Possible growth?
Many examples, no characterization
NO
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Paperfolding word
wT(P,P,)T(P)
P0?1? a pattern
0
the paperfolding word
aw(n)8n4 for n gt 13
A generalization Toeplitz words
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First results and classification
Linearar. compl.
Exponentialar. compl.
Fixed pointsof uniform morphisms
Thue-Morse word
Paperfolding word
Avgustinovich, Fon-Der-Flaass, Frid, 00 (03)
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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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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)

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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, kq.
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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
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Regular Toeplitz words
P1ab?cd? a (3-regular) pattern
P2ef? a (3-regular) pattern
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Linearity
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
  wT(P1,P2,), where
• all patterns Pi are p-regular for some fixed
  prime p
• sequence P1,P2, is ultimately periodic

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Another example
P0?1? 2-regular paperfolding
pattern
Q23? 3-regular
wQT(P,P,)T(P1,P2, P1,P2,), where
0
3
2
2
3
2
0
1
2
0
2
3
3
0
3
P12?0?3?2?1?3?
P23?0?2?3?1?2?
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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?
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Relatives of period doubling word
Let a be a symbol, p be a prime integer. Define
Rp(a)ap-1?
and
wpT(Rp(0),Rp(1),, Rp(0),Rp(1),)
0100 0101 0100 0100 0100 0101 0100 0100
period doubling word
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Minimal ar. complexity
and these limits are minimal for uniformly
recurrent words
Avgustinovich, Cassaigne, Frid, submitted
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Plot for ar. complexity of w
p
ar. compl.
length
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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
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More classification
Sturmian words,O(n3)
Exponentialar. compl.
Symm. D0L words
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Words with aw(n)O(nfu(logp(n)))
Recall that for a symbol a and a prime p
Rp(a)ap-1?.
u0010
000000001 000000001 000000000
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A theorem
Theorem. For all u (on a finite alphabet) and
each prime pgt2, aw(u)(n)O(nfu(logp(n
))).
for p2, the situation is more complicated
since 01010101... may occur both in
and
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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)

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Particular cases - 2

• If fu(n)O(an), then aw(n)O(n1log a)
  so, on the binary alphabet we can reach
  aw(n)O(n1log 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
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Geometric definition of Sturmian words
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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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Dual picture gates and faces
Berstel, Pocchiola, 93
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Counting faces
By Euler formula, fe-v1

where is the Euler function
We have
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Computational results
It seems that for 1/3lt lt2/3,
where is a simple function, ultimately
periodic on .
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The current state
linear subword complexity
Sturmian words,O(n3)
Exponentialar. compl.
Symm. D0L words
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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

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Thank you