Problems and Results related to Waring’s problem: Maximal functions and ergodic averages
We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [Mag97], [Mag02] and [MSW02]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, -maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.
In this paper we generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on a certain class of hypersurfaces. Next we apply this result to arithmetic maximal functions and ergodic averages. The approximation formula was previously known in sufficiently large dimensions while the maximal and ergodic theorems were previously known only for – see [Mag02]. Using recent bounds of Wooley, our contribution is an improved error estimate in the approximation formula and an improved range of spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces. In related papers we investigate applications to Szemerédi theorems as in [Mag08], discrepancy theory as in [Mag07] and restriction theory as in [HL11].
1.1. Arithmetic maximal functions
Fix the degree and dimension , positive integers. We define the arithmetic -sphere of radius in dimensions as
, the arithmetic -sphere of radius , contains lattice points. is possibly non-empty only when ; we denote the set of positive radii such that by . For a function and , we introduce the -spherical averages, dyadic -spherical maximal function and (full) -spherical maximal function respectively,
Throughout, all averages will be restricted to ; that is, only -spheres with lattice points on them; in particular, the dyadic supremum above is restricted to .
These maximal functions are the arithmetic analogues of continuous maximal functions over -spheres in Euclidean space. In the continuous setting, maximal functions associated to compact convex hypersurfaces are bounded on a range of spaces depending on the geometry of the hypersurface. Therefore, it is natural to ask: when is bounded on ? For sufficiently large , when is sufficiently large with respect to . Testing the maximal operator on the Dirac delta function ( is 1 if and 0 otherwise), we expect that the maximal operator is bounded on for . Let be the smallest dimension such that . By the Hardy–Littlewood circle method and Jacobi’s 4-squares formula, we know that for , see Theorem 4.1 on p. 20 of [Dav05] and Theorem 3.6 on p. 304 of [SS03] respectively. The asymptotics for when ; in particular, the value of is an open problem in number theory. Recent work of Wooley, in particular Theorem 1.4 on page 4 of [Woo12], shows that for . We will always assume that the dimension so that this holds.
1.2. Previous results and conjectures
In [Mag97], Magyar initiated the study arithmetic -spherical maximal functions and proved that the dyadic maximal operator is bounded uniformly in for a range of spaces depending on the degree and dimension.
If , then is bounded on for and . If , then is bounded on for and .
For continuous maximal functions, the boundedness of the dyadic maximal operator is equivalent to the full maximal function by the use of Littlewood–Paley theory. However, this argument fails in the arithmetic setting, and a new idea is needed to understand the full maximal function. Building on Magyar’s work, [MSW02] studied the full maximal function for degree proving the arithmetic analogue of Stein’s spherical maximal theorem – see [Ste76].
Let and , then is bounded on for .
If , then . Testing on the delta function, one deduces that is unbounded on for . For dimensions the maximal function is only bounded on . This is because in , there are precisely 24 lattice points on a sphere of radius ; that is, for all .
Subsequently, [Ion04] improved the Magyar–Stein–Wainger result by proving the restricted weak-type result at the endpoint .
Let and , then is bounded from to .
This result is analogous to the Bourgain’s restricted weak-type result for the continuous spherical maximal function in 3 or more dimensions – see [Bou85].
[Mag02] extended the results of Magyar–Stein–Wainger to positive definite, nondegenerate, homogeneous integral forms to prove boundedness of the corresponding maximal operator on and pointwise convergence of their ergodic averages when ; this includes the family of -spheres considered here. Based on these examples and results, we conjecture when is bounded on for .
If , then is bounded on for .
If , then is bounded from to .
By interpolation with the trivial bound for on , Conjecture 2 implies Conjecture 1.
1.3. Summary of results
An important and novel ingredient in Magyar–Stein–Wainger’s result is their approximation formula. We extend this to higher degrees as in [Mag02], but now take advantage of refined knowledge for exponential sums and oscillatory integrals related to -spheres.
The Approximation Formula.
Let be the characteristic function of on . If and , then for
for some .
Here and throughout the paper, is the Gauss sum of degree while is the Gelfand–Leray measure on the continuous -sphere with as its -variable Fourier transform. is a smooth function supported in and 1 in . The approximation lemma says that we can approximate the -variable Fourier transform of the arithmetic surface measure of a -sphere as a weighted sum of pieces of a localized -variable Fourier transform of the continuous -sphere with , an error term that has a power saving in the radius.
If and , then is bounded on for .
We refine this to a restricted weak-type endpoint result in a subsequent paper. In [Mag02], Magyar also investigated related ergodic theorems. We improve his results for -spheres. Suppose that we have a probability space with measure and a strongly ergodic (commuting) family of invertible measure preserving transformations. We use these transformations to define actions on and functions. For a function , define the -spherical average of radius as
If , and for some , then
for -a.e. .
The ranges and in Theorem 1 are chosen for aesthetic reasons. Theorem 3 gives a more flexible version of Theorem 1 which depends crucially on supremum bounds for a certain class of exponential sums. We phrase our bounds using a hypothesis on these exponential sums that is based on works in Waring’s problem. Then Theorem 1 is deduced by using Wooley’s sup bound, [Woo12], Theorem 1.5, p. 5. There are immediate improvements for . We discuss the best currently known results and conjectural limitations of our method in Section 5.
1.4. Structure of the paper
In section 2, we briefly mention notation used througout the paper. We hope that there will be a mix of readers from harmonic analysis, ergodic theory and number theory, so we try to make the exposition as self-contained as possible. In section 3, we discuss continuous maximal functions over hypersurfaces and derive bounds for continuous -spherical maximal functions. In section 4, we state some of the important machinery in [MSW02]; in particular we will need the Magyar–Stein–Wainger transference principle in order to exploit the machinery in section 3. In section 5, we introduce a hypothesis for exponential sums that allows us to generalize The Approximation Formula and Theorem 1. We state Wooley’s recent bounds in [Woo12]. Theorem 1 will then be an immediate application of Wooley’s bounds and Theorem 3. In section 6, we study the dyadic maximal operators. In section 7, we prove The Approximation Formula of section 5. In section 8, we combine the analysis in sections 6 and 7 to prove Theorem 3. In section 9, we use Theorem 3 to a mean ergodic theorem and our pointwise ergodic theorem for -spherical averages.
Before discussing the machinery in proving Theorem 1, we introduce some notation. Our notation will be a mix of notations from analytic number theory and harmonic analysis. Most of our notation is standard, but there are a few differences based on aesthetics.
The torus may be identified with any box in of sidelengths 1, for instance or .
will denote the character for or .
We use the non-standard notation of which we identify with the set and is the group of units in .
For , define if , or and use the dot product . Furthermore, we abuse notation by writing to mean for and the dot product notation to mean for .
For two functions , if for some constant . if for each with depending on . and are comparable if and . Finally, we may use if is much smaller than . All constants throughout the paper may depend on dimension and degree .
If , then we define its Fourier transform by for . If , then we define its Fourier transform by for . If , then we define its Fourier transform by for .
We introduce several related convolution operators such as the averaging operators , and defined on , and a measure space , respectively. The averages are intimately connected to one another and we will distinguish them by using mathcal font for an operator on , normal font for operators on and mathfrak font for operators on .
3. Estimates for continuous maximal functions over hypersurfaces
In this section we discuss the continuous analogues of our arithmetic -spherical maximal functions. There is a wide literature on continuous maximal functions over hypersurfaces. We discuss two results, one due to Bruna–Nagel–Wainger and another due to Rubio de Francia. We then apply them to deduce estimates for continuous -spherical maximal functions in Proposition 3.1. In section 7 these estimates will be applied later using the Magyar–Stein–Wainger transference principle.
3.1. Measures on hypersurfaces
There are several natural measures associated with a hypersurface . Before we can state the necessary results, we need to be precise about which surface measure we are using. We suppose that the hypersurface, , is defined by a function such that is non-singular, that is, for . We use the Gelfand–Leray measure which is defined as the unique form such that
where . The Gelfand–Leray measure is equal to the induced Lebesgue measure and the Dirac delta-measure restricted to (for definitions of these measures see [Ste93], page 498). The Gelfand–Leray measure is also equal to an appropriately normalized Euclidean surface measure – see Proposition 2 in [Mag02]. However, most important for us is the distributional description of the Gelfand–Leray measure given now.
If is a Schwartz function on and is a Schwartz function on , then
In particular, this is true if we take . The proof of this fact follows from a change of variables and the Fourier inversion theorem. For the details of the proof, see Lemma 2 in [Mag02].
3.2. bounds for maximal functions over hypersurfaces
Let be the sphere of radius in with surface area measure normalized to have for all . For a continuous function with compact support, define the spherical average of radius by
and the spherical maximal function
E. Stein was the first to investigate the spherical maximal function, proving boundedness on the sharp range of spaces when .
In dimensions , the spherical maximal function is bounded on when .
The spherical maximal function is unbounded on for . This can be seen by considering the characteristic function of the unit cube, a delta mass at the origin, or the scale invariant version if and otherwise.
In this section, we are interested in generalizations of Stein’s spherical maximal theorem to hypersurfaces. Let be a smooth, convex hypersurface of finite type in for and be its Gelfand–Leray measure; we say that a hypersurface is convex if the body it bounds is convex, and a hypersurface is finite type if at each point , all tangent lines at make finite order of contact. Furthermore, let be a smooth, positive, compactly supported function on ( is arbitrary, but all implicit constants below may depend on it). Similarly to the spherical maximal function, define the averages and maximal function respectively by
Nontrivial estimates for the maximal function occur when the surface satisfies a curvature condition such as everywhere positive Gaussian curvature in the case of the sphere or the finite type condition. Such a curvature condition is reflected in the decay of the Fourier transform of the surface measure which implies that the associated maximal function is bounded on a range of spaces. A surface can fail to have nontrivial bounds when the surface is very flat – see [CM86] for more details.
Below we choose two exemplary theorems to prove the boundedness of the continuous -spherical maximal function. The first theorem, due to Bruna–Nagel–Wainger, relates the decay estimates of the Fourier transform of a surface measure to the curvature of the surface. For , denote the outward unit normal to at by and to be the tangent plane at . For , define the -ball about as .
If is a smooth, convex hypersurface of finite type in with surface measure , then
The second theorem, due to Rubio de Francia, relates the decay estimates of the Fourier transform of a surface measure to boundedness of the maximal function on spaces.
Rubio de Francia maximal theorem.
Suppose that the dimension is at least 3. If for all and some , then is a bounded operator on for .
Since , the range of extends below 2. In particular, is bounded on .
3.3. bounds for maximal functions over -spheres
We now combine the Bruna–Nagel–Wainger and Rubio de Francia theorems to establish bounds for maximal operators associated to -spheres. Recall that and the hypersurface with its normalized Gelfand–Leray measure. is finite type of order , and
uniformly for . (1) is sharp at the poles, e.g. . Since is a compact surface, we may take . Thus by the Bruna–Nagel–Wainger theorem, the Fourier decay estimates are
uniformly for . Applying the Rubio de Francia theorem, we conclude:
For , is a bounded operator on if .
For the continuous maximal functions we know the sharp Fourier decay estimates. However, these do not necessarily imply the sharp maximal function estimates. In particular, the bounds for -spheres are not optimal, but they are sufficient for our applications since for . There have been many results and much progress in this area – see for instance [IKM10], but the general problem is still open.
4. Some machinery of Magyar–Stein–Wainger
In this section, we review some of the machinery in [MSW02]. In particular, we recall the transference principle of Magyar–Stein–Wainger and two inequalities that will be useful later.
4.1. The Magyar–Stein–Wainger transference principle
Given The Approximation Formula, it will be necessary to understand the relationship between multipliers defined on and . Suppose that is a multiplier supported in , then we can think of as a multiplier on or ; denote this as and respectively where is the periodization of . These have convolution operators and respectively. For ,
and for ,
Equivalently, let be the kernel of ,
and be the kernel of ,
Then is smooth, and is , the restriction of to the lattice .
We extend these notions to Banach spaces. Let be two Banach spaces, possibly infinite dimensional, with norms , and is the space of bounded linear tranformations from to . Let be the space of functions such that and be the space of functions such that . For , suppose that is a multiplier with convolution operators on and on . Extend periodically to to define with convolution operator on defined by .
Magyar–Stein–Wainger transference lemma.
The implicit constant is independent of and .
4.2. An inequality for -periodic multipliers on the torus
Lemma 4.1 (Magyar–Stein–Wainger).
Suppose that is a multiplier on where is smooth and supported in with convolution operator on . Furthermore, assume that is -periodic ( if ). For a -periodic sequence, define the Fourier transform . Then for ,
with implicit constants depending on and , but not on .
Since have disjoint supports for different , ; this gives the bound by Plancherel’s theorem. The bound follows from Minkowski’s inequality because the kernel is and since is smooth. Interpolation finishes the lemma. ∎
4.3. The main inequality
An essential ingredient in the proof of Theorem 1 is the following inequality. Let be a function on , and for , let be a convolution operator on with multiplier for a measurable subset of or , and is a measurable function on depending on , an additional parameter which is fixed for this discussion ( will vary later). By Fourier inversion, we have
This last expression does not depend on , but does depend on . Therefore,
This allows us to estimate the norm of the dyadic maximal function.
Lemma 4.2 (Main inequality).
where is the length of . In what follows, the point below will be to bound for , a major or minor arc, uniformly for in .
5. Hypothesis and the general form of Theorem 1
Our proof of Theorem 1 relies heavily on bounds for exponential sums and oscillatory integrals. The necessary bounds for oscillatory integrals were reviewed in section 3. Note that the range of spaces for the continuous -spherical maximal functions in Proposition 3.1 uses sharp bounds for oscillatory integrals and is larger than possible for the arithmetic -spherical maximal functions. Meanwhile the sharp bounds for our exponential sums are unknown. Motivated by Waring’s problem and Vinogradov’s mean value conjecture, we now describe our essential hypothesis on exponential sums which plays a similar role to the Bruna–Nagel–Wainger bounds (2) for the Fourier transform of the continuous -spherical surface measure.
Suppose that there exists integers relatively prime such that . Then
with implicit constants depending on , but independent of and .
The circle method in [Dav05] shows that if Hypothesis is true, then .
Taking , we recover the trivial bound with a logarithmic-loss. The goal is to take as large as possible. Weyl gave the first non-trivial bound showing that we can take . So the hypothesis is not vacuous. There are many important works improving Weyl’s bound, but the best asymptotic bound in is currently due to Wooley in [Woo12].
If , then is true with , and if , then is true with .
We now phrase The Approximation Formula and Theorem 1 in terms of our hypothesis and record the best results currently available.
The Approximation Formula.
Let be the characteristic function of on . If hypothesis is true for some , then for , the Fourier transform can be decomposed as
where the error term is a multiplier term with convolution operator satisfying
for some .
Fix the degree . If is true for some , then is bounded on for and .
In particular, the Weyl bound allows us to take for while Wooley’s sup bound allows us to take for . Wooley’s bounds improve on the classical Weyl bound for . It is unclear how small we can expect to take . Therefore, the -spherical maximal function is bounded on if
and for .
At the bottom of page 196 in [Mon94], Montgomery conjectures that one can take . Montgomery’s conjecture implies the -spherical maximal function is bounded on for and . Note that this is still a order of away from the conjectured endpoint because a factor of is lost by using sup bounds. In Waring’s problem the loss of a factor of by sup bounds is overcome in by using mean values of exponential sums. Regrettably, our method does not exploit this technique. We state similar improvements to the pointwise ergodic theorem in section 9.
6. The dyadic maximal operator
In this section we prove that is uniformly bounded in on for a range of -spaces depending on Hypothesis . Our analysis follows closely the analysis in [Mag97] and [AS06]; we include the proof for completeness. Theorem 4 will be used in the proof of Theorem 3 in section 8.
If Hypothesis is true for some , then for all , is uniformly bounded in on for and .
In section 6.1 we follow the circle method paradigm and decompose our operators into pieces corresponding to major and minor arcs. In section 6.2 we bound each major arc piece in Lemma 6.1, and in section 6.3 we bound each minor arc piece in Lemma 6.2. Theorem 4 follows immediately from Corollaries 6.1 and 6.2.
6.1. The circle method decomposition
If the dimension is sufficiently large, say , then . This allows us to redefine the averages to be
at the expense of a constant. is a convolution operator with multiplier where
is the Fourier transform of the characteristic function of the set of lattice points on the -sphere . Using the orthogonality relation
for , we rewrite as
for any fixed . The sum above is over lattice points in a ball of radius since the -sphere is contained in the ball of radius for any degree . Note that
where is the characteristic function of the ball of radius .
Given , define the Farey sequence of level as the set written in increasing order. The Farey sequence allows us to partition the unit interval in the following way. For each , suppose that are neighbors, and let . This does not make sense at the endpoints, 0 and 1, and we include these points by letting . Then is the disjoint union of arcs for . These are called arcs, a term from the original version of the circle method. We will need to make each arc (almost) symmetric about 0, so we shift by to get the arc . For , . Threfore, each arc has length . See [HW98] for more details on the Farey sequence and Diophantine approximation. We make a Farey dissection of level on . This decomposes into the disjoint union of arcs for and . The Farey dissection induces the following decomposition on :
We isolate each piece to define for an arc ,
By translating to , we find that
We now define the major and minor arcs; let be the major arcs and be the minor arcs. This splits the multiplier into major and minor arc pieces. Define the major arc multiplier
and minor arc multiplier
and are their respective convolution operators normalized so that and similarly for . Fix . By the triangle inequality, we reduce to bounding the dyadic maximal major arc operator and the dyadic maximal minor arc operator:
6.2. Major arcs bounds for the dyadic maximal operator
We begin our analysis of the dyadic maximal function by studying the dyadic maximal function of a major arc piece. It will be convenient to replace the exponential sum with a sharp cut-off by the smoothed exponential sum , for complex with in a major arc and . We recognize this as the Fourier transform on of an analytic function and use Poisson summation to estimate the smoothed exponential sum. This will lead us to the following lemma which is the main result of this section.
If is on a major arc and , then