Jamie Murdoch, Pure Mathematics, University of Waterloo
"Almost every set of the correct density is Λ(p)"
I
give
an
overview
of
Bourgain’s
argument
in
his
1989
paper,
Bounded
orthogonal
systems
and
the
Λ(p)-set
problem.
Through
a
rather
complex
argument,
Bourgain
was
able
to
show,
using
elementary
probabilistic
methods,
that
for
any
2
<
p
<
∞,
almost
every
random
set
is
Λ(p),
where
the
mean
density
of
the
random
sets
is
2/p.
Prior
attempts
at
this
problem
by
Rudin
in
1960
and
Hajela
in
1986
had
focused
on
the
case
where
p
is
an
even
integer,
in
which
case
the
p-norm
can
be
expanded
explicitly
using
a
multinomial
expansion,
and
sets
can
be
constructed
by
number-theoretic
and
combinatorial
arguments.
Bourgain’s
result
is
remarkable
because
it
estimates
the
p-norm
directly
for
arbitrary
p.
I
aim
to
give
an
overview
of
his
highly
technical
argument,
focusing
on
the
key
steps
where
the
mean
density
is
used.