This is the first of a series of posts concerning the Rudin-Hardy-Littlewood Conjecture. To give a taste of the problem right away let us consider to be a trigonometric polynomial of the form
where for . The main question we are interested in is whether one has an inequality of the form:
where the implied constant depends only on .
Conjecture 1 was stated in this form by Walter Rudin himself for example in [R] but the the first (and essentially only) results on this question go back to Hardy and Littlewood (see for example [R]).
Inequalities of the form (1) have deep number theoretic implications. For example, let (`S’ for squares) denote the maximum number of squares in the arithmetic progression as we vary over positive integers . Then, inequality (1) for a specific , implies that . Assuming inequality (1) for values of arbitrarily close to we would then conclude that for all we have the bound . Rudin has actually conjectured that while, at the moment, the best known bound (due to Bombieri and Zannier) is . Thus, there are two parallel conjectures, that always go hand in hand:
As we have already observed, conjecture 1 implies conjecture 2. There are also several other number-theoretic and combinatorial implications and connections that we’ll only superficially discuss here.
2. -sets and Rudin’s conjecture.
We begin by discussing Rudin’s approach from [R].
Definition 3 Let . A function is called an -function if and whenever . A trigonometric polynomial which is an -function is called an -polynomial. We will denote by the space of all -functions that belong to .
In order to define the notion of -sets we need the following simple observation:
- (i) .
- (ii) .
Proof: It is obvious that (i) implies (ii). To see that (ii) implies (i) we can interpolate by writing so that
which in turns implies (i).
In other words the property for only depends on the larger index. This allows us to define -sets as follows:
for all -polynomials .
- (i) If then will be called a -set if for all -polynomials .
- (ii) If then will be called a -set if for all -polynomials .
Of course the -property makes sense for but we won’t discuss this here.
2.1. Equivalent formulations
Proposition 6 Let . Then is a -set if and only if for all .
Proof: Assume first that is a -set. Obviously it is enough to show that . Assuming that we see that the Cesáro means of are -polynomials in with norms uniformly bounded by the norm of . Now by the property of the set the Cesáro means are -polynomials which are uniformly in . We conclude that and of course is an -function so we are done.
To prove the other direction just observe that if , then and are two norms in the same Banach space and must therefore be equivalent.
The -property is essentially a restriction phenomenon and that is better illustrated by the following reformulation of the problem. For a set let us consider the restriction operator acting initially on trigonometric polynomials in the following way:
or where is the indicator function of the set .
The definition of the property together with the fact that is a self-adjoint operator gives the following equivalent characterization:
- Let . Then is a -set if and only if extends to a bounded operator from to . By duality this is equivalent to being bounded from to .
2.2. Arithmetic Progressions and -sets
As we have mentioned in the Introduction, the -property of a subset of the integers was considered in connection to the problem of studying how many elements of we can find in arithmetic progressions of length . For a positive integer let us define to be the number of terms which has in the arithmetic progression
positive integers and .
Theorem 8 Suppose that is a -set for some , that is if for all -polynomials we have
Proof: We have two proofs of the theorem. We begin with the one due to Rudin in [R]. Let be the arithmetic progression and suppose that . Observe that by definition . The `natural choice’ of the -polynomial to use with Rudin’s conjecture is
whose norm we can easily control by the -property of . Indeed, for any function , where is the dual exponent of , we have
where in the before-last inequality we have used the -property of and the fact that is an -polynomial. On the other hand we have that
In order to make these estimates useful we need to find a test function whose Fourier coefficients we can easily control. There are several choices here that are possible but let us work with the Fejér kernel in the place of . For the Fejér kernel, , we have that and . Interpolating between and , , , we get
This is the desired control of the -norm of our test function. How about its Fourier coefficients? Well, since for , if all the coefficients we were considering were in the range then we would be done. This however is not necessarily the case since we are calculating Fourier coefficients corresponding to some frequencies in the set . We can mend this situation by dilating the Fejér kernel and suitably translating its frequencies. Indeed, observe that the function satisfies whenever . Now, defining where , we have that
for some . For any such we have that so we conclude that for all . Now we have fixed the Fourier coefficients of the function but what about its -norm? It is easy to see that this hasn’t changed due to the fact that is an integer. Putting all the estimates together we conclude
A similar albeit more elegant way to prove this relies on Proposition 7. Let us define . A standard calculation shows that for all . Assuming that is a -set for some we get from Proposition 7 tha is a bounded operator from to . This means that
However, has as many distinct frequencies as the members of , that is , so that . We conclude that .
2.3. Rudin’s conjecure on the set of squares
be the set of squares. Then Conjecture 1 reads:
Conjecture 9 (Rudin’s Conjecture) The set of squares is a -set for all .
Some remarks are in order. First of all the conjecture is open (to the best of my knowledge) for any but the interesting number theoretic implications happen only in the range . On the other hand, the set of squares is not a -set so the restriction is best possible. This was first observed by Rudin in [R]. We repeat the proof of this fact here using a different argument.
Proposition 10 The set of squares is not a -set.
Proof: We consider the trigonometric polynomial which is obviously an -polynomial. Obviously . On the other hand, we have that
Now, for a positive integer, let be the number of representations of as a sum of two squares of positive integers
Then observe that we have
Taking for granted the classical number theoretic asymptotic estimate
we conclude that
which shows in particular that is not a -set.
- [HL]. Hardy, G. H., Littlewood, J. E., Some problems of diophantine approximation. Acta Math. 37 (1914), no. 1, 193–239.
- [R]. W. Rudin, Trigonometric Series with Gaps, Indiana Univ. Math. J. 9 No. 2 (1960), 203–227.