1. A theorem on the growth of exponential sums evaluated at polynomials
Let be a positive integer and fix . We are looking at bounds for
where is a large number. The main theorem in this regard is the following.
Let . Then
for all outside a set of measure tending to 0 with N.
2. A probabilistic model
We will prove such theorem by showing that for any the above exponential sum will have size bigger than only on a set of measure tending to with .
Indeed, suppose that is now being chosen at random in , where the space is equipped with the uniform probability on . Let
By expanding the square and using the formula
if and otherwise,
we may easily deduce that
where is the set of positive integers solution to the system of equations given by
This number is actually usually called Vinogradov’s mean value and denoted by . Its analytical representation is indeed
By a well known theorem of Bourgain, Demeter and Guth we have for any and
However, here the situation is much easier and we do not need to appeal to such an important theorem. Indeed, since , the system has only trivial solutions. More precisely, once fixed a solution all the other solutions are just a rearrangement of this tuple. One can see this by noticing that the solutions to the system
with positive integers determine via Newton’s identities all the elementary symmetric functions in . Thus any other solution of it will just be a permutation of the roots of the polynomial We conclude that in such case
Thus, since we get Since , which is equivalent to , by Chebyshev’s inequality we get
which leads to
Taking now and by the arbitrariness of we deduce that for all and all , apart from those in a certain set of measure at most , tending to with .
3. Some further considerations
Remark 1 Considering higher moments of is it possible to prove for instance that
or equivalently, deviates a bit from its mean value.
Moreover, as we also have , as convergence in law.
Some natural questions now arises:
Can the above procedures be made effective?
Can one do better by considering different scales?
Regarding the last one, we just point out that the random variables behave like a random walk. Therefore, under some further assumptions one could use the iterated law of logarithms to deduce that
Acknowledgement: The above discussion is a slightly enlarged version of what was presented by Sam Chow during one of the usual Number Theory meetings at Warwick.