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 .
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 .
If is a sequence of -valued functions on that converges in to a -valued function , then .
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.
Let be the set of functions for of density at least Then is dense in
Let . We construct a sequence of functions that converge to We use duality and Fourier transform. We have and
For , 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 .