COLLOQUIUM
Roots of 2, and the continued
fraction of the minimal asymptotic
dispersion
Professor Mojtaba Moniri
Western Illinois University
Abstract: The optimal modulus of convergence of the roots of
2 to their limit 1 (where the ever decreasing tolerances above 1
are of the form 1/n for input positive integers n) is a curious
subsequence of two inhomogeneous Beatty sequences
superimposed. Both candidate patterns have slope 1/ln(2) but
their intercepts are +/- 1/2, and the agreement is dominantly
with the lower possibility. The doubles 2/(2^(1/n) -1) break
the intervals (2n/ln(2) -1, 2n/ln(2)) into two subintervals, and
usually it is the right one which captures an integer and for
such n's the mentioned dominance prevails. But sometimes the
trapped integer is in the left subinterval, the first time being at
n=777451915729368. If, as it happens for this n and the next
two, that integer (2243252046704766 in this case) is even, an
exceptional modulus value is achieved. Otherwise the double is
unusual but n is not, the smallest such n being
3052446177238342414. The slope 1/ln(2) (or sometimes its
half depending on adoption) is the least asymptotic dispersion
among all sequences in the unit interval, realized by the base-2
logs of dyadic rationals in the interval [1,2]. The continued
fraction of this number yields consequences on term selection
above. Mathematica with MaxExtraPrecision is useful to
confirm, among other things, the extra jump for the less
assumed type of values rather deep into initial segments.
Department of
Mathematics
Thursday,
March 27, 2014
3:45 p.m.
204 Morgan Hall
Refreshments will be
served at 3:30 p.m.