Polynomial sequences in discrete nilpotent groups of step 2

1.1 The main theorem

Our main theorem in this article concerns L p boundedness of maximal averages along polynomial sequences in discrete nilpotent groups of step 2, L p pointwise ergodic theorems, and L 2 boundedness of singular integrals. More precisely:

The theorem follows by combining the main results in [35] for parts (i) and (ii) and [37] for part (iii). We discuss now some connections between this theorem and other related results in the literature.

1.2 The universal step-two group G 0

The proof of Theorem 1.1 will follow from our second main result, Theorem 1.3, for averages on universal nilpotent groups of step two. We start with some definitions. For integers d ≥ 1 , we define

and the “universal” step-two nilpotent Lie groups G 0 # = G 0 # ( d )

with the group multiplication law

Alternatively, we can also define the group G 0 # as the set of elements

where d ′ ≔ d ( d − 1 ) ∕ 2 and Y d ′ ≔ { ( l 1 , l 2 ) ∈ Y d : l 2 ≥ 1 } . Letting

we notice that the product rule in the group G 0 # is given by

if g = ( g ( 1 ) , g ( 2 ) ) and h = ( h ( 1 ) , h ( 2 ) ) . For any g = ( g ( 1 ) , g ( 2 ) ) ∈ G 0 # , its inverse is given by

The second variable of g = ( g ( 1 ) , g ( 2 ) ) ∈ G 0 # is called the central variable. Based on the product structure (1.9) of the group G 0 # , it is not difficult to see that g ⋅ h = h ⋅ g for any g = ( g ( 1 ) , g ( 2 ) ) ∈ G 0 # and h = ( 0 , h ( 2 ) ) ∈ G 0 # .

Let G 0 = G 0 ( d ) denote the discrete subgroup

Let A 0 : R → G 0 # denote the canonical polynomial map (or the moment curve on G 0 # )

and notice that A 0 ( Z ) ⊆ G 0 . For x = ( x l 1 l 2 ) ( l 1 , l 2 ) ∈ Y d ∈ G 0 # and Λ ∈ ( 0 , ∞ ) , we define

Notice that the dilations Λ ∘ are group homomorphisms on the group G 0 that are compatible with the map A 0 , i.e., Λ ∘ A 0 ( x ) = A 0 ( Λ x ) .

Let χ : R → [ 0 , 1 ] be a smooth function supported on the interval [ − 2 , 2 ] . Given any real number N ≥ 1 and a function f : G 0 → C , we can define a smoothed average along the moment curve A 0 by the formula

The main advantage of working on the group G 0 with the polynomial map A 0 is the presence of the compatible dilations Λ ∘ defined in (1.12), which lead to a natural family of associated balls. This can be efficiently exploited by noting that M N χ is a convolution operator on G 0 .

The convolution of functions on the group G 0 is defined by the formula

Then it is not difficult to see that M N χ ( f ) ( x ) = f * G N χ ( x ) , where

We are now ready to state our second main result.

1.3 Remarks and overview of the proof

We discuss now some of the main ideas in the proofs of Theorems 1.1 and 1.3.

1.4 The main difficulty and a nilpotent circle method

Bourgain’s seminal articles [11–13] generated a large amount of research and progress in the field. Many other discrete operators have been analyzed by many authors motivated by problems in analysis and ergodic theory. See, for example, [15,36,38–41,44–48,49,50,52] for some results of this type and more references. A common feature of all of these results, which plays a crucial role in the proofs, is that one can use Fourier analysis techniques, in particular, the powerful framework of the classical circle method, to perform the analysis.

Our situation in Theorem 1.3 is different. The main conceptual issue is that there is no good Fourier transform on nilpotent groups, compatible with the structure of the underlying convolution operators and at the level of analytical precision of the classical circle method. At a more technical level, there is no good resolution of the delta function compatible with the group multiplication on the group G 0 . This prevents us from using a naive implementation of the circle method. The classical delta function resolution

does not detect the group multiplication correctly. Here, ( y ( 1 ) − x ( 1 ) ) . θ ( 1 ) and ( y ( 2 ) − x ( 2 ) ) . θ ( 2 ) denote the usual scalar product of vectors in R d and R d ′ , respectively.

These issues lead to very significant difficulties in the proof and require substantial new ideas. Our main new construction in [35] is what we call a nilpotent circle method, an iterative procedure, starting from the center of the group and moving down along its central series. At every stage, we identify “minor arcs,” and bound their contributions using Weyl’s inequalities (the classical Weyl inequality as well as a nilpotent Weyl inequality which was proved in [37]). The final stage involves “major arcs” analysis, which relies on a combination of continuous harmonic analysis on groups G 0 # and arithmetic harmonic analysis over finite integer rings modulo Q ∈ Z + . We outline this procedure in Section 3.

At the implementation level, classical Fourier techniques are replaced with almost orthogonality methods based on exploiting high order T ∗ T arguments for operators defined on the discrete group G 0 . Investigating high powers of T ∗ T (i.e., ( T ∗ T ) r for a large r ∈ Z + ) is consistent with a general heuristic lying behind the proof of Waring-type problems, which says that the more variables that occur in Waring-type equations, the easier it is to find solutions, and we are able to make this heuristic rigorous in our problem. Manipulating the parameter r , by taking r to be very large, we can always decide how many variables we have at our disposal, making our operators “smoother and smoother.”

1.5 General discrete nilpotent groups

The primary goal is, of course, to remove the restriction that the discrete nilpotent groups G in Theorem 1.1 are of step 2 and thus establish the full Conjecture 1.2 in the linear m = 1 case for arbitrary invertible measure-preserving transformations T 1 , … , T d that generate a nilpotent group of any step k ≥ 2 . The iterative argument we outline in Section 3 below could, in principle, be extended to higher step groups, at least as long as the group and the polynomial sequence have suitable “universal”-type structure, as one could try to go down along the central series of the group and prove minor arcs and transition estimates at every stage.

However, this is only possible if one can prove suitable analog of the nilpotent Weyl’s inequalities in Proposition 2.1 on general nilpotent groups of step k ≥ 3 . The point is to have a small (not necessarily optimal, but nontrivial) gain for bounds on oscillatory sums over many variables, corresponding to the kernels of high power ( T ∗ T ) r operators, whenever frequencies are restricted to the minor arcs. In our case, the formulas are explicit, see the identities (2.10), and we can use ideas of Birch [9] and Davenport [21] for Diophantine forms in many variables to control the induced oscillatory sums, but the analysis seems to be more complicated for the higher step nilpotent groups.

1.6 Waring-type problems

The classical Waring problem, solved by Hilbert [30] in 1909, concerns the possibility of writing any positive integer as a sum of finitely many p powers: for any integer p ≥ 1 , there is r = r ( p ) such that any integer y ∈ Z + can be written in the form

There is a vast amount of literature on this problem and its many possible extensions. In particular, the symmetric system of equations

first studied by Vinogradov [55] in relation to the Waring problem, have been the focus of intense recent research, see [58] for some recent results.

We are interested here in understanding the analogs question on our discrete nilpotent Lie group G 0 , and for our given polynomial sequence A 0 : Can one represent elements g ∈ G 0 in the form

for some integers n 1 , m 1 , … , n r , m r , provided that r is large enough? We are, in fact, interested in proving a quantitative statement on the number of such representations, for integers n 1 , m 1 , … , n r , m r ∈ [ N ] ≔ [ − N , N ] ∩ Z .

We notice that many group elements g cannot be written in the form (1.23), due to local obstructions; for instance, if g can be represented in the form (1.23) then necessarily g 10 ≡ g 20 ≡ … ≡ g d 0 ( mod 2 ) , g 10 = g 30 = … ( mod 3 ) etc. We remark also there is a significant difference between the classical Waring problem (1.21) and its nilpotent analog (1.23), namely the positivity of the p -powers which imposes size restrictions on the variables x i in terms of the prescribed output value y .

For integers r , N ≥ 1 and g ∈ G 0 let

A qualitative variant of the Waring problem on nilpotent groups was recently investigated in [33,34], see also the references given there. Our main result in this direction is the following quantitative version on the nilpotent group G 0 :

We will provide a complete proof of this theorem in Section 4.

1.7 Organization

The rest of this article is organized as follows: in Section 2, we present several nilpotent Weyl estimates proved in [37], which play a key role in the analysis of minor arcs. In Section 3, we outline our main new method, the nilpotent circle method, developed in [35] to prove maximal and variational estimates on nilpotent groups. In Section 4, we prove Theorem 1.4, the main new result in this article.

2.1 High-order T ∗ T arguments and product kernels

Many of our ℓ 2 ( G 0 ) estimates will be based on high-order T ∗ T arguments. Assume that

are convolution operators defined by some ℓ 1 ( G 0 ) kernels L 1 , K 1 , … , L r , K r : G 0 → C , i.e., S j f = f ∗ L j and T j f = f ∗ K j for j ∈ { 1 , … , r } . Then the adjoint operators S 1 ∗ , … , S r ∗ are also convolution operators, defined by the kernels L 1 ∗ , … , L r ∗ given by

Moreover, by using (1.14), for any f ∈ ℓ 2 ( G 0 ) and x ∈ G 0 , we have

In other words ( S 1 ∗ T 1 … S r ∗ T r f ) ( x ) = ( f ∗ A r ) ( x ) , where the kernel A r is given by

To use these formulas, we decompose h j = ( h j ( 1 ) , h j ( 2 ) ) , g j = ( g j ( 1 ) , g j ( 2 ) ) as in (1.7). Then

as a consequence of applying (1.9) inductively.

In many of our applications, the operators S 1 , T 1 , … , S r , T r are equal and, more importantly, are defined by a kernel K that has product structure, i.e.,