The discrete spherical averages over a family of sparse sequences

In [Ste76], Stein introduced the spherical maximal function and proved that it is bounded on $L^{p}(\mathbb{R}^{d})$ for $p>\frac{d}{d-1}$ and $d\geq 3$. This was later extended to $p>2$ when $d=2$ by Bourgain in [Bou86]. Recently, discrete analogues of Stein’s spherical maximal function have been considered. The discrete sphere of radius $r\geq 0$ in $\mathbb{Z}^{d}$ is $S^{d-1}(r):=\{x\in\mathbb{Z}^{d}:|x|^{2}=r^{2}\}$ which contains $N_{d}(r)=\#S^{d-1}(r)$ lattice points. For dimensions $d\geq 4$, the set $S^{d-1}(r)$ is non-empty precisely when $r^{2}\in\mathbb{N}$. Let $\mathcal{R}_{full}$ denote the set of radii $r$ such that $S^{d-1}(r)\not=\emptyset$, then $\mathcal{R}_{full}$ is precisely $\left\{r\in\mathbb{R}_{\geq 0}:r^{2}\in\mathbb{N}\right\}$ when $d\geq 4$. For $r\in\mathcal{R}_{full}$, we introduce the discrete spherical averages:

where $\sigma_{r}:=\frac{1}{N_{d}(r)}{\bf 1}_{\{x\in\mathbb{Z}^{d}:|x|^{2}=r^{2}\}}$ is the uniform probability measure on $S^{d-1}(r)$. The associated (full) maximal function is

Motivated by Stein’s theorem, it is natural to ask: when is $A_{*}$ bounded on $\ell^{p}(\mathbb{Z}^{d})$? Testing the maximal operator on the delta function and using the asymptotics for the number of lattice points on spheres, $N_{d}(r)\eqsim r^{d-2}$ when $d\geq 5$, we expect that the maximal operator is bounded on $\ell^{p}$ for $p>\frac{d}{d-2}$ when $d\geq 5$. In fact, building on the work of [Mag97], this was proven in [MSW02] with a subsequent restricted weak-type bound at the endpoint $p=\frac{d}{d-2}$ proven in [Ion04]. In particular, $A_{*}$ is a bounded operator from $\ell^{p,1}(\mathbb{Z}^{d})$ to restricted $\ell^{p,\infty}(\mathbb{Z}^{d})$ for $p=\frac{d}{d-2}$; that is, $A_{*}$ is restricted weak-type $(\frac{d}{d-2},\frac{d}{d-2})$. This result is sharp. For generalizations to higher degree varieties where the sharp ranges of $\ell^{p,\infty}(\mathbb{Z}^{d})$ are unknown, we refer the reader to [Mag02] and [Hug13].

Shortly after Stein’s work on the spherical maximal function [Ste76], it was observed by Calderón and Coifman–Weiss that lacunary versions of Stein’s spherical maximal function are bounded on a larger range of $L^{p}(\mathbb{R}^{d})$-spaces than for the full Stein spherical maximal function – see [Cal79] and [CW78] respectively. In particular, they proved:

Similarly, we define the lacunary discrete spherical maximal function when $d\geq 5$ by restricting the set of radii to lie in a lacunary sequence $\mathcal{R}:=\{r_{j}\}_{j\in\mathbb{N}}\subset\mathcal{R}_{full}$. Recall that a sequence is lacunary if $r_{j+1}>c\,r_{j}$ for some $c>1$. More generally, for any $\mathcal{R}\subseteq\mathcal{R}_{full}$, the discrete spherical maximal function over $\mathcal{R}$ is defined in the natural way as

By the Magyar–Stein–Wainger discrete spherical maximal theorem in [MSW02], we know that any discrete spherical maximal function in 5 or more dimensions over a subsequence of radii in $\mathcal{R}_{full}$ is bounded on $\ell^{p}(\mathbb{Z}^{d})$ for $d\geq 5$ and $p>\frac{d}{d-2}$. In particular this holds true for any lacunary subsequence in 5 or more dimensions.

It is conjectured that the continuous lacunary spherical maximal function is bounded from $L^{1}(\mathbb{R}^{d})$ to $L^{1,\infty}(\mathbb{R}^{d})$ for $d\geq 2$. See [STW03a] and [STW03b] for recent work in this direction. Analogously, it is a folklore conjecture that the arithmetic lacunary spherical maximal function is bounded on $\ell^{p}(\mathbb{Z}^{d})$ for $p>1$. The following conjecture is our motivation for this paper.

Surprisingly, J. Zienkiewicz has shown that Conjecture 1 is false in general.¹¹1This counterexample was communicated to the author by Zienkiewicz after an initial draft of this paper was completed. More precisely, Zienkiewicz proved that there exist infinite, yet arbitrarily thin subsets $\mathcal{R}\subset\mathcal{R}_{full}$ such that $A_{\mathcal{R}}^{*}f$ is unbounded on $\ell^{p}(\mathbb{Z}^{d})$ for $1\leq p<\frac{d}{d-1}$ and $d\geq 5$. Zienkiewicz’s counterexamples proceed by a probabilistic argument that incorporates information about the discrete spherical averages when one reduces mod $Q$ for $Q\in\mathbb{N}$. By a probabilistic argument, Zienkiewicz constructs counterexamples that violate (5) of G below for infinitely many primes. In Section 6 we revise Conjecture 1 to account for these counterexamples.

Our main theorem is the following improvement to the range of boundedess for maximal functions over lacunary sequences of radii possessing an interesting dichotomy.

Theorem 1 reduces our problem to finding sequences of natural numbers satisfying certain arithmetic properties, and it would be superfluous if we could not find a sequence of radii satisfying the G and B dichotomy. Our next theorem gives a family of sequences satisfying these conditions. This family is well known in number theory as it includes primorials, also known as Euclidean primes, whose definition is motivated by Euclid’s proof of the infinitude of primes. For these sequences, (5) is simple to verify. However, (6) is difficult to verify, and we only have very poor bound for it in this article. In turn, for our family of sparse sequences, presently we are only able to show that the associated discrete spherical maximal function is strong-type at the Magyar–Stein–Wainger endpoint for the full discrete spherical maximal function as opposed to restricted weak-type bound in [Ion04].

Theorem 1 and Theorem 2 immediately combine to yield the following endpoint Magyar–Stein–Wainger theorem applied to such sequences.

An intriguing aspect of Corollary 1.1 is that $A_{\mathcal{R}}^{*}$ is bounded on $\ell^{2}(\mathbb{Z}^{4})$. This is surprising since the full discrete spherical maximal function, $A_{*}$ fails to be bounded on $\ell^{2}(\mathbb{Z}^{4})$. Worse yet, for dimensions $d\leq 4$, the full maximal function is only bounded on $\ell^{\infty}(\mathbb{Z}^{d})$. Theorem 1 and Corollary 1.1 mark the first results in 4 dimensions for boundedness of the arithmetic spherical maximal function over infinite sequences.

Let us examine the four dimensional situation further. In $\mathbb{Z}^{4}$, there are precisely 24 lattice points on a sphere of radius $2^{j}$ for all $j\in\mathbb{N}$, e.g. $N_{4}(2^{j})=24$. Applying the discrete spherical maximal function to the delta function demonstrates that the naive definition of our maximal function in 4 dimensions is wrong. However, further considerations suggest that there could be a version of the Magyar–Stein–Wainger theorem in 4 dimensions. To make this precise, we must account for some arithmetic phenomena. From the work of Hardy–Littlewood on the circle method, we have the asymptotic formula

where $\mathfrak{S}(r^{2})$ is the singular series, which satisfies $\mathfrak{S}(r^{2})\eqsim 1$ when $d\geq 5$. Lagrange’s theorem and Jacobi’s four square theorem demonstrate that the 4 dimensional case (i.e. $S^{3}(r)\in\mathbb{Z}^{4}$) is different. In four dimensions, the bound for the error term in (7) dominates the main term, and therefore, (7) is not useful as an asymptotic. However, Kloosterman was able to refine their method by exploiting oscillation between Gauss sums to improve (7) to

for all $\epsilon>0$ and $d\geq 4$. The cost here is that the singular series in the asymptotic formula is not uniform, and in fact can be very small. One can predict this from Jacobi’s theorem since there are precisely 24 lattice points when $r=2^{j}$ for all $j\in\mathbb{N}$; in this case, one sees that $\mathfrak{S}(4^{j})\lesssim 4^{-j}$. To avoid this 2-adic obstruction in the singular series when $d=4$, we make the additional assumption that $r_{j}^{2}\not\equiv 0(\mod 4)$ for each $j\in\mathbb{N}$ or (4) holds for the prime 2. In either case $N_{4}(r^{2})\eqsim r^{2}$ so that there are many lattice points on $S^{3}(r)$. Modifying the discrete spherical maximal function in 4 dimensions in this way, it is natural to conjecture that it is bounded on $\ell^{p}(\mathbb{Z}^{4})$ for $2<p\leq\infty$ – see [Hug12] for a precise statement of this conjecture and a related result.

Our notation is a mix of notations from analytic number theory and harmonic analysis. Most of our notation is standard, but there are a few choices based on aesthetics.

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  The torus $\mathbb{T}^{d}$ may be identified with any box in $\mathbb{R}^{d}$ of sidelengths 1, for instance $[0,1]^{d}$ or $[-1/2,1/2]^{d}$.

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  We identify $\mathbb{Z}/q\mathbb{Z}$ with the set $\left\{1,\dots,q\right\}$ and $(\mathbb{Z}/q\mathbb{Z})^{\times}$ is the group of units in $\mathbb{Z}/q\mathbb{Z}$, also considered as a subset of $\left\{1,\dots,q\right\}$.

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  $e\left(t\right)$ will denote the character $e^{2\pi it}$ for $t\in\mathbb{R},\mathbb{Z}/q\mathbb{Z}$ or $\mathbb{T}$.

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  We abuse notation by writing ${b}^{2}$ to mean $\sum_{i=1}^{d}b_{i}^{2}$ for $b\in(\mathbb{Z}/q\mathbb{Z})^{d}$ and the dot product notation $b\cdot m$ to mean $\sum_{i=1}^{d}b_{i}m_{i}$ for $b,m\in(\mathbb{Z}/q\mathbb{Z})^{d}$ or $\mathbb{Z}^{d}$.

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  For any $q\in\mathbb{N}$, $\varphi(q)$ will denote Euler’s totient function, the size of $(\mathbb{Z}/q\mathbb{Z})^{\times}$.

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  For two functions $f,g$, $f\lesssim g$ if $\lvert f(x)\rvert\leq C\lvert g(x)\rvert$ for some constant $C>0$. $f$ and $g$ are comparable $f\eqsim g$ if $f\lesssim g$ and $g\lesssim f$. All constants throughout the paper may depend on dimension $d$.

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  If $f:\mathbb{R}^{d}\to\mathbb{C}$, then we define its Fourier transform by $\widetilde{f}(\xi):=\int_{\mathbb{R}^{d}}f(x)e(x\cdot\xi)dx$ for $\xi\in\mathbb{R}^{d}$. If $f:\mathbb{T}^{d}\to\mathbb{C}$, then we define its Fourier transform by $\widehat{f}(m):=\int_{\mathbb{T}^{d}}f(x)e(-m\cdot x)dx$ for $m\in\mathbb{Z}^{d}$. If $f:\mathbb{Z}^{d}\to\mathbb{C}$, then we define its inverse Fourier transform by $\widehat{f}(\xi):=\sum_{m\in\mathbb{Z}^{d}}f(m)e(n\cdot\xi)$ for $\xi\in\mathbb{T}^{d}$.

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  $\left\Arrowvert T\right\Arrowvert_{p\to p}$ will denote the $\ell^{p}(\mathbb{Z}^{d})$ to $\ell^{p}(\mathbb{Z}^{d})$ operator norm of the operator $T$.

By interpolation with the usual $\ell^{\infty}(\mathbb{Z}^{d})$ bound, we restrict our attention to the range $1\leq p\leq 2$. From [MSW02], we understand that each average decomposes into a main term (resembling the singular series and singular integral of the circle method) and an error term. We recall this machinery in section 2. The main term and error term will be bounded on ranges of $\ell^{p}(\mathbb{Z}^{d})$-spaces by distinct arguments. In section 3, our bounds for the main term exploit the Weil bounds for Kloosterman sums via the transference principle of Magyar–Stein–Wainger. The main result here is Lemma 3.1 and we introduce a more precise decomposition of the multipliers in order to use the Kloosterman method. In section 4, the error term is handled by a square function argument using the lacunary condition. The main lemma here is Lemma 4.2. Our novelty here is that we exploit (well-known) cancellation for averages of Ramanujan sums to improve the straight-forward $\ell^{1}(\mathbb{Z}^{d})$ bound. Theorem 1 follows immediately by combining Lemma 3.1 with Lemma 4.2. In section 5, we prove Theorem 2. The properties of our sequences are well known to analytic number theorists, but we could not find them in the literature. Section 6 concludes our paper with some questions and remarks.