Ostrowski Numeration

In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers.

Fix a positive irrational number ''α'' with continued fraction expansion 'a''₀; ''a''₁, ''a''₂, ... Let (''q''_''n'') be the sequence of denominators of the convergents ''p''_''n''/''q''_''n'' to α: so ''q''_''n'' = ''a''_''n''''q''_''n''−1 + ''q''_''n''−2. Let ''α''_''n'' denote ''T''^''n''(''α'') where ''T'' is the Gauss map ''T''(''x'') = , and write ''β''_''n'' = (−1)^''n''+1 ''α''₀ ''α''₁ ... ''α''_''n'': we have ''β''_''n'' = ''a''_''n''''β''_''n''−1 + ''β''_''n''−2.

Real number representations

Every positive real ''x'' can be written as

:$x = \sum_^\infty b_n \beta_n \$

where the integer coefficients 0 ≤ ''b''_''n'' ≤ ''a''_''n'' and if ''b''_''n'' = ''a''_''n'' then ''b''_''n''−1 = 0.

Integer representations

Every positive integer ''N'' can be written uniquely as

:$N = \sum_^k b_n q_n \$

where the integer coefficients 0 ≤ ''b''_''n'' ≤ ''a''_''n'' and if ''b''_''n'' = ''a''_''n'' then ''b''_''n''−1 = 0.

If ''α'' is the golden ratio, then all the partial quotients ''a''_''n'' are equal to 1, the denominators ''q''_''n'' are the Fibonacci numbers and we recover Zeckendorf's theorem on the Fibonacci representation of positive integers as a sum of distinct non-consecutive Fibonacci numbers.

See also

* Complete sequence

References

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* {{cite book , last=Pytheas Fogg , first=N. , editor1=Berthé, Valérie, editor1-link=Valérie Berthé, editor2=Ferenczi, Sébastien, editor3=Mauduit, Christian, editor4=Siegel, Anne , title=Substitutions in dynamics, arithmetics and combinatorics , series=Lecture Notes in Mathematics , volume=1794 , location=Berlin , publisher=Springer-Verlag , year=2002 , isbn=3-540-44141-7 , zbl=1014.11015

Non-standard positional numeral systems