Numbers normal to one base but not to another base
Dave L. Renfro
[sci.math Tue, 02 Jul 2002 10:05:13 +1000]
http://mathforum.org/epigone/sci.math/punsisum
wrote
> Considerably more is known. Given any two multiplicatively
> independent integers (that is, no power of the one is a power
> of the other) there's a continuum of reals normal to the one
> base and not normal to the other. I think that's a result of
> Wolfgang Schmidt.
>
> I think it goes even farther - given any two (finite?) sets of
> integers, no product of powers of elements of the one equal to
> a product of powers of elements of the other, etc., etc.
Since I've been working on an essay about exceptional sets and
normal numbers, I happen to have some of this at my fingertips.
Cassels [1] proved there exist c-many real numbers that are normal
with respect to every integer that is not a power of 3 and yet whose
base-3 expansion omits the digit 2 (i.e. they fail to be normal with
respect to base 3 in a very strong way).
[1] J.W.S. Cassels, "On a problem of Steinhaus about normal numbers",
Colloq. Math. 7 (1959) 95-101. [MR 22 #4694; Zbl 47.04402]
Let r and s be integers larger than 1. We say that r and s are
multiplicatively independent if r^m = s^n for some integers m and n,
and multiplicatively dependent otherwise. It is not difficult to
prove that if r and s are multiplicatively dependent, then any
number that is normal to base r must also be normal to base s.
Schmidt [2] proved that if r and s are multiplicatively independent,
then there exist c-many real numbers that are normal to base r but
which are not even simply normal to base s. In fact, there will be
c-many such real numbers in every nonempty open interval, since any
finite initial segment of digits can be altered at will without
affecting any of these normality properties.
[2] Wolfgang M. Schmidt, "On normal numbers", Pacific J. Math. 10
(1960), 661-672. [MR 22 #7994; Zbl 93.05401]
[3] Wolfgang M. Schmidt, "Über die Normalität von Zahlen zu
verschiedenen Basen" [On normality with different bases],
Acta Arith. 7 (1961/62), 299-309. [MR 25 #3902; Zbl 105.26702]
In Schmidt [3] this result was strengthened as much as possible in
one direction. Suppose we partition of {2, 3, 4, ...} into two
disjoint sets R and S such that if b_1 and b_2 are multiplicatively
dependent, then b_1 and b_2 belong to the same set. [We will say that
the pair (R,S) is *multiplicatively independent* when this condition
holds.] Schmidt [3] proved that there exist c-many real numbers x such
that x is normal to each base in R and x is not normal to each base
in S. For the same reason I mentioned above, we actually have c-many
such numbers in every nonempty open interval.
[[ I haven't seen a copy of this paper (and besides, it's
in German and I can't read German), so I don't know
whether Schmidt [3] actually proved the numbers are not
simply normal to each base in S or if he just proves the
weaker result that each of the numbers is not normal to
each base in S. The Mathematical Review for Schmidt [3]
only states the weaker version. Other books and papers
I've looked at which mention Schmidt [3] also only state
the weaker version. Does anyone happen to know if Schmidt
proved in [3] that each of the c-many numbers that are normal
to every base in R fails to be SIMPLY normal to every base
in S? (I know that being simply normal to each positive
integer power of b implies normality to base b, but this
is the wrong direction for what I'm asking about.) ]]
Note that by simply letting R = {r^n: n = 1, 2, ...} and S be
the complement of R relative to {2, 3, ...}, then (since s will
belong to S) we already have a stronger result than what is proved
in Schmidt [2], at least modulo the "not even simply normal" issue
discussed in the indented and bracketed paragraph above.
Nagasaka [4] proved the following result: If r and s are
multiplicatively independent, then the set of real numbers that
are normal to base r and not normal to base s has Hausdorff
dimension 1. Besides strengthening one version of Schmidt [2]
(the version where "not even simply normal" is weakened to "not
normal"), this also strengthens a result due to Beyer [5], namely
that the non-normal numbers have Hausdorff dimension 1. [Actually,
Beyer (his theorem 3 on pp. 39-40) proved a result that is
superficially weaker in one way and a bit stronger in another way:
The set of real numbers x such that the limiting frequency of either
the digit '0' or the digit '1' does NOT exist in the base-2 expansion
of x has Hausdorff dimension 1. (The property of not being simply
normal to base 2 means that either the limiting frequency of one of
the digits doesn't exist (because we're dealing with base 2, this is
equivalent to neither digit having a limiting frequency), or that
one of the limiting frequencies exists and isn't equal to 1/2.)]
Among other things, Nagasaka [6] proved that the set of all
non-normal numbers and the set of all simply normal but not
normal numbers each has Hausdorff dimension 1.
Schmidt [7] proved that the set of numbers not normal to any
base has Hausdorff dimension 1.
Lars Olsen (see [8]) has recently proved some interesting results
concerning a very extreme form of non-normality. We say that a real
number x is *extremely non-normal* if, for each base b = 2, 3, ...,
every real number in [0,1] is a subsequence limit for the limiting
frequency of every finite block of digits from {0, 1, ..., b}. Let E
be the set of extremely non-normal numbers. Then E is small in the
sense of Hausdorff dimension (E has Hausdorff dimension 0) but large
in the sense of Baire category and large in the sense of packing
dimension (E is co-meager and E has packing dimension 1). Regarding
the set E, see also my item #11 at
http://mathforum.org/epigone/sci.math/yeplaxdum/getei6...@forum.mathforum.com
For some remarks about an even more extreme type of non-normality,
see my sci.math post at
http://mathforum.org/epigone/sci.math/khanspeutwa/28ae5e5e.02060...@posting.google.com
Since any finite initial segment of digits can be altered at will
without affecting any of these normality properties, all of these
Hausdorff dimension 1 properties are actually "everywhere of Hausdorff
dimension 1" (i.e. in every nonempty open interval the collection
of numbers having the property has Hausdorff dimension 1). Similarly
for Olsen's packing dimension 1 result. [Incidentally, Olsen gets
the packing dimension 1 result as a consequence of the fact that
any subset of the reals with packing dimension less than 1 must
be a first category set.]
[4] Kenji Nagasaka, "La dimension de Hausdorff de certaines
ensembles dans [0,1]" [The Hausdorff dimension of certain sets
in [0,1]], Proc. Japan Acad. Ser. A, Math. Sci. 54 (1979),
109-112. [MR 58 #16580; Zbl 403.10030]
[5] William A. Beyer, "Hausdorff dimension of level sets of some
Rademacher series", Pacific J. Math. 12 (1962), 35-46.
[MR 25 #4063; Zbl 154.05401]
[6] Kenji Nagasaka, "On Hausdorff dimension of non-normal sets",
Ann. Inst. Statist. Math. 23 (1971), 515-521.
[MR 48 #2103; Zbl 255.10052]
[7] Wolfgang M. Schmidt, "On badly approximable numbers and certain
games", Trans. Amer. Math. Soc. 123 (1966), 178-199.
[MR MR 33 #3793; Zbl 163.04802]
[8] Lars Olsen, "Extremely non-normal numbers", preprint, 2002,
8 pages.
Getting back to the matter of normality with respect to one base
but not with respect to another base, Pollington [9] proved a
Hausdorff dimension 1 strengthening of Schmidt [3] --->>>
Given any partition of {2, 3, ...} into a pair of multiplicatively
independent sets R and S, the set of real numbers x such that x is
normal to every base in R and x is not normal to every base in S
has Hausdorff dimension 1.
[9] Andrew D. Pollington, "The Hausdorff dimension of a set of
normal numbers", Pacific J. Math. 95 (1981), 193-204.
[MR 83k:10098; Zbl 479.10031]
I'll end with a discussion of something that's somewhat related to
the type of result that Cassels [1] proved. Given a positive integer
b > 1, we say that a real number is disjunctive to base b if its
b-ary expansion contains, to the right of its decimal point, every
finite string of digits from {0, 1, ..., b-1}. Equivalently, we can
insist that every finite string of digits from {0, 1, ..., b-1}
occurs infinitely often. [For example, if every finite string of
digits occurs at least once, to see that '458' occurs infinitely
often note that we must have the strings '458', '458458', '458458458',
'458458458458', etc.] What can we say about numbers that are
disjunctive relative to one base but which are not disjunctive
relative to another base?
It can be shown that if a number x is disjunctive to base r, then
x is disjunctive to any base that is multiplicatively dependent to r.
Hence, we need to deal with multiplicatively independent bases.
Let r and s are multiplicatively independent bases, and let
S = {S_1, S_2, ..., S_n} be a finite collection of finite digit
sequences from {0, 1, ..., s-1}. Let N(S) be the set of real numbers
whose s-ary expansion does not contain any of the digit sequences
belonging to S. Note that N(S) might not contain any irrational
numbers. For example, if s = 10 and S = {0, 1, 2, 4, ..., 9}, then
1/3 is the only number in N(S). However, El-Zanati/Bransue [10]
proved that if N(S) contains an irrational number, then N(S) contains
a number that is disjunctive to base r.
Let r and s be multiplicatively independent bases. Then Hertling [11]
proved that the set of real numbers x such that x is disjunctive
to base r and not disjunctive to base s has cardinality c. Hertling
also gave some explicit examples. For instance, the sum from k=0
to infinity of s^(-k! - k) is not disjunctive to base s, but it is
disjunctive to every base r such that r is multiplicatively
independent
of s and r is divisible by every prime factor of s.
For other results along these lines, see Section 5 (pp. 13-20) of
Calude/Priese/Staiger [12].
[10] Saad I. El-Zanati and William R.R. Transue, "On dynamics
of certain Cantor sets", J. Number Theory 36 (1990), 246-253.
[MR 91i:11009; Zbl 719.11010]
[11] Peter Hertling, "Disjunctive omega-words and real numbers",
The Journal of Universal Computer Science 2 (1996), 549-568.
[MR 2000e:68097; Zbl 955.68065]
[12] Cristian S. Calude, Lutz Priese, and Ludwig Staiger,
"Disjunctive Sequences: An Overview", CDMTCS-063 (October 1997),
Centre for Discrete Mathematics and Theoretical Computer
Science, University of Auckland, New Zealand, 39 pages.
An 871 K .pdf file can be found at
http://www.cs.auckland.ac.nz/staff-cgi-bin/mjd/secondcgi.pl
Dave L. Renfro