On simply normal numbers to different bases
Let s be an integer greater than or equal to 2 . A real number is simply normal to base s if in its base- s expansion every digit 0 , 1 , … , s - 1 occurs with the same frequency 1 / s . Let S be the set of positive integers that are not perfect powers, hence S is the set { 2 , 3 , 5 , 6 , 7 , 10...
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Published in | Mathematische annalen Vol. 364; no. 1-2; pp. 125 - 150 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.02.2016
Springer Verlag |
Subjects | |
Online Access | Get full text |
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Summary: | Let
s
be an integer greater than or equal to
2
. A real number is simply normal to base
s
if in its base-
s
expansion every digit
0
,
1
,
…
,
s
-
1
occurs with the same frequency
1
/
s
. Let
S
be the set of positive integers that are not perfect powers, hence
S
is the set
{
2
,
3
,
5
,
6
,
7
,
10
,
11
,
…
}
. Let
M
be a function from
S
to sets of positive integers such that, for each
s
in
S
, if
m
is in
M
(
s
)
then each divisor of
m
is in
M
(
s
)
and if
M
(
s
)
is infinite then it is equal to the set of all positive integers. These conditions on
M
are necessary for there to be a real number which is simply normal to exactly the bases
s
m
such that
s
is in
S
and
m
is in
M
(
s
)
. We show these conditions are also sufficient and further establish that the set of real numbers that satisfy them has full Hausdorff dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to different bases. |
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ISSN: | 0025-5831 1432-1807 |
DOI: | 10.1007/s00208-015-1209-9 |