<s>
An	O
ordinary	O
fractal	B-Language
string	I-Language
is	O
a	O
bounded	O
,	O
open	O
subset	O
of	O
the	O
real	O
number	O
line	O
.	O
</s>
<s>
Such	O
a	O
subset	O
can	O
be	O
written	O
as	O
an	O
at-most-countable	O
union	O
of	O
connected	O
open	O
intervals	O
with	O
associated	O
lengths	O
written	O
in	O
non-increasing	O
order	O
;	O
we	O
also	O
refer	O
to	O
as	O
a	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
For	O
example	O
,	O
is	O
a	O
fractal	B-Language
string	I-Language
corresponding	O
to	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
A	O
fractal	B-Language
string	I-Language
is	O
the	O
analogue	O
of	O
a	O
one-dimensional	O
"	O
fractal	B-Application
drum	O
,	O
"	O
and	O
typically	O
the	O
set	O
has	O
a	O
boundary	O
which	O
corresponds	O
to	O
a	O
fractal	B-Application
such	O
as	O
the	O
Cantor	O
set	O
.	O
</s>
<s>
The	O
heuristic	O
idea	O
of	O
a	O
fractal	B-Language
string	I-Language
is	O
to	O
study	O
a	O
(	O
one-dimensional	O
)	O
fractal	B-Application
using	O
the	O
"	O
space	O
around	O
the	O
fractal.	O
"	O
</s>
<s>
It	O
turns	O
out	O
that	O
the	O
sequence	O
of	O
lengths	O
of	O
the	O
set	O
itself	O
is	O
"	O
intrinsic	O
,	O
"	O
in	O
the	O
sense	O
that	O
the	O
fractal	B-Language
string	I-Language
itself	O
(	O
independent	O
of	O
a	O
specific	O
geometric	O
realization	O
of	O
these	O
lengths	O
as	O
corresponding	O
to	O
a	O
choice	O
of	O
set	O
)	O
contains	O
information	O
about	O
the	O
fractal	B-Application
to	O
which	O
it	O
corresponds	O
.	O
</s>
<s>
For	O
each	O
fractal	B-Language
string	I-Language
,	O
we	O
can	O
associate	O
to	O
a	O
geometric	O
zeta	O
function	O
:	O
the	O
Dirichlet	O
series	O
.	O
</s>
<s>
Informally	O
,	O
the	O
geometric	O
zeta	O
function	O
carries	O
geometric	O
information	O
about	O
the	O
underlying	O
fractal	B-Application
,	O
particularly	O
in	O
the	O
location	O
of	O
its	O
poles	O
and	O
the	O
residues	O
of	O
the	O
zeta	O
function	O
at	O
these	O
poles	O
.	O
</s>
<s>
These	O
poles	O
of	O
(	O
the	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
)	O
the	O
geometric	O
zeta	O
function	O
are	O
then	O
called	O
complex	O
dimensions	O
of	O
the	O
fractal	B-Language
string	I-Language
,	O
and	O
these	O
complex	O
dimensions	O
appear	O
in	O
formulae	O
which	O
describe	O
the	O
geometry	O
of	O
the	O
fractal	B-Application
.	O
</s>
<s>
For	O
fractal	B-Language
strings	I-Language
associated	O
with	O
sets	O
like	O
Cantor	O
sets	O
,	O
formed	O
from	O
deleted	O
intervals	O
that	O
are	O
rational	O
powers	O
of	O
a	O
fundamental	O
length	O
,	O
the	O
complex	O
dimensions	O
appear	O
in	O
an	O
arithmetic	O
progression	O
parallel	O
to	O
the	O
imaginary	O
axis	O
,	O
and	O
are	O
called	O
lattice	O
fractal	B-Language
strings	I-Language
(	O
For	O
example	O
,	O
the	O
complex	O
dimensions	O
of	O
the	O
Cantor	O
set	O
are	O
,	O
which	O
are	O
an	O
arithmetic	O
progression	O
in	O
the	O
direction	O
of	O
the	O
imaginary	O
axis	O
)	O
.	O
</s>
<s>
In	O
fact	O
,	O
an	O
ordinary	O
fractal	B-Language
string	I-Language
is	O
Minkowski	O
measurable	O
if	O
and	O
only	O
if	O
it	O
is	O
non-lattice	O
.	O
</s>
<s>
A	O
generalized	O
fractal	B-Language
string	I-Language
is	O
defined	O
to	O
be	O
a	O
local	O
positive	O
or	O
complex	O
measure	O
on	O
such	O
that	O
for	O
some	O
,	O
where	O
the	O
positive	O
measure	O
is	O
the	O
total	O
variation	O
measure	O
associated	O
to	O
.	O
</s>
<s>
These	O
generalized	O
fractal	B-Language
strings	I-Language
allow	O
for	O
lengths	O
to	O
be	O
given	O
non-integer	O
multiplicities	O
(	O
among	O
other	O
possibilities	O
)	O
,	O
and	O
each	O
ordinary	O
fractal	B-Language
string	I-Language
can	O
be	O
associated	O
with	O
a	O
measure	O
that	O
makes	O
it	O
into	O
a	O
generalized	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
An	O
ordinary	O
fractal	B-Language
string	I-Language
is	O
a	O
bounded	O
,	O
open	O
subset	O
of	O
the	O
real	O
number	O
line	O
.	O
</s>
<s>
We	O
refer	O
to	O
as	O
a	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
The	O
fractal	B-Language
string	I-Language
of	O
the	O
Cantor	O
set	O
is	O
called	O
the	O
Cantor	O
string	O
.	O
</s>
<s>
The	O
geometric	O
information	O
of	O
the	O
Cantor	O
set	O
in	O
the	O
example	O
above	O
is	O
contained	O
in	O
the	O
ordinary	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
This	O
notion	O
of	O
fractal	B-Application
dimension	O
can	O
be	O
generalized	O
to	O
that	O
of	O
complex	O
dimension	O
,	O
which	O
may	O
be	O
used	O
to	O
deduce	O
geometrical	O
information	O
regarding	O
the	O
local	O
oscillations	O
in	O
the	O
geometry	O
of	O
the	O
fractal	B-Application
.	O
</s>
<s>
For	O
example	O
,	O
the	O
complex	O
dimensions	O
of	O
a	O
fractal	B-Language
string	I-Language
(	O
such	O
as	O
the	O
Cantor	O
string	O
)	O
may	O
be	O
used	O
to	O
write	O
an	O
explicit	O
tube	O
formula	O
for	O
the	O
volume	O
of	O
an	O
-neighborhood	O
of	O
the	O
fractal	B-Language
string	I-Language
,	O
and	O
the	O
presence	O
of	O
non-real	O
complex	O
dimensions	O
corresponds	O
to	O
oscillatory	O
terms	O
in	O
this	O
expansion	O
.	O
</s>
<s>
For	O
each	O
fractal	B-Language
string	I-Language
,	O
we	O
can	O
associate	O
to	O
a	O
geometric	O
zeta	O
function	O
defined	O
as	O
the	O
Dirichlet	O
series	O
.	O
</s>
<s>
Poles	O
of	O
the	O
geometric	O
zeta	O
function	O
are	O
called	O
complex	O
dimensions	O
of	O
the	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
The	O
general	O
philosophy	O
of	O
the	O
theory	O
of	O
complex	O
dimensions	O
for	O
fractal	B-Language
strings	I-Language
is	O
that	O
complex	O
dimensions	O
describe	O
the	O
intrinsic	O
oscillation	O
in	O
the	O
geometry	O
,	O
spectra	O
and	O
dynamics	O
of	O
the	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
For	O
a	O
fractal	B-Language
string	I-Language
with	O
infinitely	O
many	O
nonzero	O
lengths	O
,	O
the	O
abscissa	O
of	O
convergence	O
coincides	O
with	O
the	O
Minkowski	O
dimension	O
of	O
the	O
boundary	O
of	O
the	O
string	O
,	O
.	O
</s>
<s>
For	O
a	O
fractal	B-Language
string	I-Language
,	O
composed	O
of	O
an	O
infinite	O
sequence	O
of	O
lengths	O
,	O
the	O
complex	O
dimensions	O
of	O
the	O
fractal	B-Language
string	I-Language
are	O
the	O
poles	O
of	O
the	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
the	O
geometric	O
zeta	O
function	O
associated	O
with	O
the	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
(	O
When	O
the	O
analytic	B-Algorithm
continuation	I-Algorithm
of	O
a	O
geometric	O
zeta	O
function	O
is	O
not	O
defined	O
to	O
all	O
of	O
the	O
complex	O
plane	O
,	O
we	O
take	O
a	O
subset	O
of	O
the	O
complex	O
plane	O
called	O
the	O
"	O
window	O
"	O
,	O
and	O
look	O
for	O
the	O
"	O
visible	O
"	O
complex	O
dimensions	O
that	O
exist	O
within	O
that	O
window	O
.	O
)	O
</s>
<s>
Continuing	O
with	O
the	O
example	O
of	O
the	O
fractal	B-Language
string	I-Language
associated	O
to	O
the	O
middle	O
thirds	O
Cantor	O
set	O
,	O
we	O
compute	O
.	O
</s>
<s>
Ordinary	O
and	O
generalized	O
fractal	B-Language
strings	I-Language
may	O
be	O
used	O
to	O
study	O
the	O
geometry	O
of	O
a	O
(	O
one-dimensional	O
)	O
fractal	B-Application
,	O
as	O
well	O
as	O
to	O
relate	O
the	O
geometry	O
of	O
the	O
object	O
to	O
its	O
spectrum	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
geometric	O
zeta	O
function	O
associated	O
to	O
a	O
fractal	B-Language
string	I-Language
may	O
be	O
used	O
to	O
write	O
an	O
explicit	O
tube	O
formula	O
for	O
the	O
volume	O
of	O
a	O
neighborhood	O
of	O
the	O
fractal	B-Application
.	O
</s>
<s>
Regarding	O
the	O
connection	O
between	O
geometry	O
and	O
spectra	O
,	O
the	O
spectral	O
zeta	O
function	O
of	O
a	O
fractal	B-Language
string	I-Language
,	O
which	O
is	O
the	O
geometric	O
zeta	O
function	O
times	O
the	O
Riemann	O
zeta	O
function	O
,	O
may	O
be	O
used	O
to	O
write	O
explicit	O
formulae	O
which	O
describe	O
spectral	O
counting	O
functions	O
.	O
</s>
<s>
The	O
framework	O
of	O
fractal	B-Language
strings	I-Language
also	O
serves	O
to	O
unify	O
aspects	O
of	O
fractal	B-Application
and	O
arithmetic	O
geometry	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
general	O
explicit	O
formula	O
for	O
counting	O
the	O
(	O
reciprocal	O
)	O
lengths	O
of	O
a	O
fractal	B-Language
string	I-Language
may	O
be	O
used	O
to	O
prove	O
Riemann	O
's	O
explicit	O
formula	O
when	O
using	O
a	O
suitable	O
generalized	O
fractal	B-Language
string	I-Language
which	O
is	O
supported	O
on	O
the	O
prime	O
powers	O
with	O
multiplicities	O
of	O
each	O
given	O
by	O
the	O
logarithm	O
of	O
the	O
prime	O
base	O
of	O
the	O
power	O
.	O
</s>
<s>
For	O
fractal	B-Language
strings	I-Language
associated	O
with	O
sets	O
like	O
Cantor	O
sets	O
,	O
formed	O
from	O
deleted	O
intervals	O
that	O
are	O
rational	O
powers	O
of	O
a	O
fundamental	O
length	O
,	O
the	O
complex	O
dimensions	O
appear	O
in	O
a	O
regular	O
,	O
arithmetic	O
progression	O
parallel	O
to	O
the	O
imaginary	O
axis	O
,	O
and	O
are	O
called	O
lattice	O
fractal	B-Language
strings	I-Language
.	O
</s>
<s>
There	O
is	O
a	O
dichotomy	O
in	O
the	O
theory	O
of	O
measures	O
of	O
such	O
objects	O
:	O
an	O
ordinary	O
fractal	B-Language
string	I-Language
is	O
Minkowski	O
measurable	O
if	O
and	O
only	O
if	O
it	O
is	O
non-lattice	O
.	O
</s>
<s>
The	O
existence	O
of	O
non-real	O
complex	O
dimensions	O
with	O
positive	O
real	O
part	O
has	O
been	O
proposed	O
by	O
Michel	O
Lapidus	O
and	O
Machiel	O
van	O
Frankenhuijsen	O
to	O
be	O
the	O
signature	O
feature	O
of	O
fractal	B-Application
objects	O
.	O
</s>
<s>
This	O
new	O
definition	O
of	O
fractality	O
solves	O
some	O
old	O
problems	O
in	O
fractal	B-Application
geometry	O
.	O
</s>
<s>
For	O
example	O
,	O
according	O
to	O
the	O
proposed	O
definition	O
of	O
fractality	O
in	O
the	O
sense	O
of	O
Mandelbrot	O
,	O
Cantor	B-Algorithm
's	I-Algorithm
devil	I-Algorithm
's	I-Algorithm
staircase	I-Algorithm
not	O
fractal	B-Application
because	O
its	O
Hausdorff	O
and	O
topological	O
dimensions	O
coincide	O
.	O
</s>
<s>
However	O
,	O
the	O
Cantor	B-Algorithm
staircase	I-Algorithm
function	O
possesses	O
many	O
features	O
which	O
ought	O
to	O
be	O
considered	O
fractal	B-Application
such	O
as	O
self-similarity	O
,	O
and	O
in	O
this	O
new	O
sense	O
of	O
fractality	O
the	O
Cantor	B-Algorithm
staircase	I-Algorithm
function	O
is	O
considered	O
fractal	B-Application
since	O
it	O
has	O
non-real	O
complex	O
dimensions	O
.	O
</s>
<s>
A	O
generalized	O
fractal	B-Language
string	I-Language
is	O
defined	O
to	O
be	O
a	O
local	O
positive	O
or	O
local	O
complex	O
measure	O
on	O
such	O
that	O
for	O
some	O
,	O
where	O
the	O
positive	O
measure	O
is	O
the	O
total	O
variation	O
measure	O
associated	O
to	O
.	O
</s>
<s>
A	O
generalized	O
fractal	B-Language
string	I-Language
allows	O
for	O
a	O
fractal	B-Language
string	I-Language
to	O
have	O
a	O
given	O
set	O
of	O
lengths	O
with	O
non-integer	O
multiplicities	O
,	O
or	O
for	O
a	O
fractal	B-Language
string	I-Language
to	O
have	O
a	O
continuum	O
of	O
lengths	O
instead	O
of	O
discrete	O
.	O
</s>
<s>
By	O
convention	O
,	O
a	O
generalized	O
fractal	B-Language
string	I-Language
is	O
supported	O
on	O
reciprocal	O
lengths	O
as	O
opposed	O
to	O
an	O
ordinary	O
fractal	B-Language
string	I-Language
which	O
is	O
a	O
multiset	O
of	O
(	O
decreasing	O
or	O
non-increasing	O
)	O
lengths	O
.	O
</s>
<s>
In	O
light	O
of	O
this	O
,	O
the	O
condition	O
that	O
the	O
measure	O
has	O
"	O
no	O
mass	O
near	O
zero	O
,	O
"	O
or	O
more	O
precisely	O
that	O
there	O
exists	O
a	O
positive	O
number	O
such	O
that	O
the	O
interval	O
has	O
measure	O
zero	O
with	O
respect	O
to	O
,	O
may	O
be	O
seen	O
as	O
an	O
analogue	O
of	O
the	O
boundedness	O
of	O
the	O
ordinary	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
an	O
ordinary	O
fractal	B-Language
string	I-Language
with	O
multiplicities	O
,	O
then	O
the	O
measure	O
associated	O
to	O
(	O
where	O
refers	O
to	O
the	O
Dirac	O
delta	O
measure	O
concentrated	O
at	O
the	O
point	O
)	O
is	O
an	O
example	O
of	O
a	O
generalized	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
Note	O
that	O
the	O
delta	O
functions	O
are	O
supported	O
on	O
the	O
singleton	O
sets	O
corresponding	O
to	O
the	O
reciprocals	O
of	O
the	O
lengths	O
of	O
the	O
ordinary	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
If	O
the	O
multiplicities	O
are	O
not	O
positive	O
integers	O
,	O
then	O
is	O
a	O
generalized	O
fractal	B-Language
string	I-Language
which	O
cannot	O
be	O
realized	O
as	O
an	O
ordinary	O
fractal	B-Language
string	I-Language
.	O
</s>
<s>
A	O
concrete	O
example	O
of	O
such	O
a	O
generalized	O
fractal	B-Language
string	I-Language
would	O
be	O
the	O
generalized	O
Cantor	O
string	O
for	O
.	O
</s>
