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Given a compact abelian group , endowed with a Haar probability measure
, let
be the set of (Borel) measurable functions
satisfying
. The average of
is called its density.
Let be system of linear forms, i.e
and
where for any
and
. For a measurable function
, one may define the
-count of
as
Here by a slight abuse of notation, the measure on
denotes in fact the product measure
of the Haar measure on
.
Let
When a system is fixed, we may omit it from the notation
. Given a sequence
of finite groups of cardinality tending to infinity, it is natural to consider the sequence
. When the infimum is taken only over
-valued measurable functions of density at least
, this is the minimum number of APs a set
of density at least
can have. But these infimum are actually the same (basically by convexity).
The case was introduced by Croot. He proved that the minimal density
of three-term arithmetic progressions converged as
through the primes. Sisask, in his PhD thesis, was able to express this limit as an integral over the corresponding limit object, namely the circle
, thus
As customary, the corresponding problem in finite field models is cleaner. Thus we set . Whereas Croot noticed that the sequence
was in some sense approximately decreasing, it is easy to see that for any system
, the sequence
is genuinely non-increasing. This is because
embeds naturally in
; thus if
is a map from
to
, it can be extended into
by putting
.
Now , and
. This proves that
, which implies that this sequence has a limit.
Let be now fixed. We want to find an appealing expression for the limit of the sequence
. It is thus tempting to look for a natural topological compact group that would play the role of limit object of the sequence
, just as
played the role of limit object for the sequence
.
The analogy leads one to introducing
and
. The group
equipped with the product topology is a compact group by Tychonoff’s theorem, and
equipped with the discrete topology is locally compact; it is the Pontryagin dual of
. It is now reasonable to ask whether
One easily notices that for any
by extending, as above, a function
into a function
on
with the same density and the same
-count. Indeed, one can define
and then is a measurable function which has same density and same
-count.
We prove the other inequality. For this we prove two intermediate propositions. We write from henceforth .
Proposition 1
If is a sequence of
-valued functions on
that converges in
to a
-valued function
, then
.
Proof
Let . We show that
converges in
to 0, which will conclude. In fact we prove a stronger convergence, a convergence in
. We show that
. For that we simply remark that
Now, any non-trivial linear map , i.e. any map of the form
with , preserves the Haar measure, which implies that
Using this, the triangle equality and the fact that the functions are 1-bounded, we conclude.
Proposition 2
Let be the set of functions
for
of density at least
Then
is dense in
Proof
Let . We construct a sequence of functions
that converge to
We use duality and Fourier transform. We have
and
For , write
Then write
This defines a function , which obviously satisfies
.
The same holds for its extension . Notice that
is also the conditional expectation with respect to the
-algebra
. More precisely,
as both equal .
This implies in particular that , and thus
, has its values in
.
To check that , we use the conditional expectation interpretation
Now we check the convergence. It follows from basic harmonic analysis: being in
, its Fourier series
converges to
in
. This concludes the proof of Proposition 2.
Thus even for 4-APs, this shows that
exists and equals
This is in contrast to a paper of Candela-Szegedy, for the same pattern (4-APs) but with
where they obtain as a limit a much more complicated infimum, an infimum over functions
on
of the
-count, where
is another pattern. The part that holds in the setting of
and not in the setting of
is Proposition 2. The truncated Fourier series of
, which is the natural
approximation, does not give a function on cyclic groups
with the same density and AP-count as
.
We remark that the paper of Hatami-Hatami-Hirst implies that the sequence of functions admits a limit object
which is defined on a finite cartesian power of
.
In fact, if is the
-APs, pattern of complexity
, and
, the limit object
is defined on
. In particular, if
, this means that the infimum above is attained, i.e.
for some measurable. When
, it is still not known whether such a result (i.e. the existence of a limit object
) holds. However, Szegedy proves the existence of a limit object
on a non-explicit group
.
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