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I’ve just uploaded to the arXiv my paper The Ionescu-Wainger multiplier theorem and the adeles“. This paper revisits a useful multiplier theorem of Ionescu and Wainger on “major arc” Fourier multiplier operators on the integers {{\bf Z}} (or lattices {{\bf Z}^d}), and strengthens the bounds while also interpreting it from the viewpoint of the adelic integers {{\bf A}_{\bf Z}} (which were also used in my recent paper with Krause and Mirek).

For simplicity let us just work in one dimension. Any smooth function {m: {\bf R}/{\bf Z} \rightarrow {\bf C}} then defines a discrete Fourier multiplier operator {T_m: \ell^p({\bf Z}) \rightarrow \ell^p({\bf Z})} for any {1 \leq p \leq \infty} by the formula

\displaystyle  {\mathcal F}_{\bf Z} T_m f(\xi) =: m(\xi) {\mathcal F}_{\bf Z} f(\xi)

where {{\mathcal F}_{\bf Z} f(\xi) := \sum_{n \in {\bf Z}} f(n) e(n \xi)} is the Fourier transform on {{\bf Z}}; similarly, any test function {m: {\bf R} \rightarrow {\bf C}} defines a continuous Fourier multiplier operator {T_m: L^p({\bf R}) \rightarrow L^p({\bf R})} by the formula

\displaystyle  {\mathcal F}_{\bf R} T_m f(\xi) := m(\xi) {\mathcal F}_{\bf R} f(\xi)

where {{\mathcal F}_{\bf R} f(\xi) := \int_{\bf R} f(x) e(x \xi)\ dx}. In both cases we refer to {m} as the symbol of the multiplier operator {T_m}.

We will be interested in discrete Fourier multiplier operators whose symbols are supported on a finite union of arcs. One way to construct such operators is by “folding” continuous Fourier multiplier operators into various target frequencies. To make this folding operation precise, given any continuous Fourier multiplier operator {T_m: L^p({\bf R}) \rightarrow L^p({\bf R})}, and any frequency {\alpha \in {\bf R}/{\bf Z}}, we define the discrete Fourier multiplier operator {T_{m;\alpha}: \ell^p({\bf Z}) \rightarrow \ell^p({\bf Z})} for any frequency shift {\alpha \in {\bf R}/{\bf Z}} by the formula

\displaystyle  {\mathcal F}_{\bf Z} T_{m,\alpha} f(\xi) := \sum_{\theta \in {\bf R}: \xi = \alpha + \theta} m(\theta) {\mathcal F}_{\bf Z} f(\xi)

or equivalently

\displaystyle  T_{m;\alpha} f(n) = \int_{\bf R} m(\theta) {\mathcal F}_{\bf Z} f(\alpha+\theta) e( n(\alpha+\theta) )\ d\theta.

More generally, given any finite set {\Sigma \subset {\bf R}/{\bf Z}}, we can form a multifrequency projection operator {T_{m;\Sigma}} on {\ell^p({\bf Z})} by the formula

\displaystyle  T_{m;\Sigma} := \sum_{\alpha \in \Sigma} T_{m;\alpha}

thus

\displaystyle  T_{m;\alpha} f(n) = \sum_{\alpha \in \Sigma} \int_{\bf R} m(\theta) {\mathcal F}_{\bf Z} f(\alpha+\theta) e( n(\alpha+\theta) )\ d\theta.

This construction gives discrete Fourier multiplier operators whose symbol can be localised to a finite union of arcs. For instance, if {m: {\bf R} \rightarrow {\bf C}} is supported on {[-\varepsilon,\varepsilon]}, then {T_{m;\Sigma}} is a Fourier multiplier whose symbol is supported on the set {\bigcup_{\alpha \in \Sigma} \alpha + [-\varepsilon,\varepsilon]}.

There are a body of results relating the {\ell^p({\bf Z})} theory of discrete Fourier multiplier operators such as {T_{m;\alpha}} or {T_{m;\Sigma}} with the {L^p({\bf R})} theory of their continuous counterparts. For instance we have the basic result of Magyar, Stein, and Wainger:

Proposition 1 (Magyar-Stein-Wainger sampling principle) Let {1 \leq p \leq \infty} and {\alpha \in {\bf R}/{\bf Z}}.
  • (i) If {m: {\bf R} \rightarrow {\bf C}} is a smooth function supported in {[-1/2,1/2]}, then {\|T_{m;\alpha}\|_{B(\ell^p({\bf Z}))} \lesssim \|T_m\|_{B(L^p({\bf R}))}}, where {B(V)} denotes the operator norm of an operator {T: V \rightarrow V}.
  • (ii) More generally, if {m: {\bf R} \rightarrow {\bf C}} is a smooth function supported in {[-1/2Q,1/2Q]} for some natural number {Q}, then {\|T_{m;\alpha + \frac{1}{Q}{\bf Z}/{\bf Z}}\|_{B(\ell^p({\bf Z}))} \lesssim \|T_m\|_{B(L^p({\bf R}))}}.

When {p=2} the implied constant in these bounds can be set to equal {1}. In the paper of Magyar, Stein, and Wainger it was posed as an open problem as to whether this is the case for other {p}; in an appendix to this paper I show that the answer is negative if {p} is sufficiently close to {1} or {\infty}, but I do not know the full answer to this question.

This proposition allows one to get a good multiplier theory for symbols supported near cyclic groups {\frac{1}{Q}{\bf Z}/{\bf Z}}; for instance it shows that a discrete Fourier multiplier with symbol {\sum_{\alpha \in \frac{1}{Q}{\bf Z}/{\bf Z}} \phi(Q(\xi-\alpha))} for a fixed test function {\phi} is bounded on {\ell^p({\bf Z})}, uniformly in {p} and {Q}. For many applications in discrete harmonic analysis, one would similarly like a good multiplier theory for symbols supported in “major arc” sets such as

\displaystyle  \bigcup_{q=1}^N \bigcup_{\alpha \in \frac{1}{q}{\bf Z}/{\bf Z}} \alpha + [-\varepsilon,\varepsilon] \ \ \ \ \ (1)

and in particular to get a good Littlewood-Paley theory adapted to major arcs. (This is particularly the case when trying to control “true complexity zero” expressions for which the minor arc contributions can be shown to be negligible; my recent paper with Krause and Mirek is focused on expressions of this type.) At present we do not have a good multiplier theory that is directly adapted to the classical major arc set (1) (though I do not know of rigorous negative results that show that such a theory is not possible); however, Ionescu and Wainger were able to obtain a useful substitute theory in which (1) was replaced by a somewhat larger set that had better multiplier behaviour. Starting with a finite collection {S} of pairwise coprime natural numbers, and a natural number {k}, one can form the major arc type set

\displaystyle  \bigcup_{\alpha \in \Sigma_{\leq k}} \alpha + [-\varepsilon,\varepsilon] \ \ \ \ \ (2)

where {\Sigma_{\leq k} \subset {\bf R}/{\bf Z}} consists of all rational points in the unit circle of the form {\frac{a}{Q} \mod 1} where {Q} is the product of at most {k} elements from {S} and {a} is an integer. For suitable choices of {S} and {k} not too large, one can make this set (2) contain the set (1) while still having a somewhat controlled size (very roughly speaking, one chooses {S} to consist of (small powers of) large primes between {N^\rho} and {N} for some small constant {\rho>0}, together with something like the product of all the primes up to {N^\rho} (raised to suitable powers)).

In the regime where {k} is fixed and {\varepsilon} is small, there is a good theory:

Theorem 2 (Ionescu-Wainger theorem, rough version) If {p} is an even integer or the dual of an even integer, and {m: {\bf R} \rightarrow {\bf C}} is supported on {[-\varepsilon,\varepsilon]} for a sufficiently small {\varepsilon > 0}, then

\displaystyle  \|T_{m;\Sigma_{\leq k}}\|_{B(\ell^p({\bf Z}))} \lesssim_{p, k} (\log(1+|S|))^{O_k(1)} \|T_m\|_{B(L^p({\bf R}))}.

There is a more explicit description of how small {\varepsilon} needs to be for this theorem to work (roughly speaking, it is not much more than what is needed for all the arcs {\alpha + [-\varepsilon,\varepsilon]} in (2) to be disjoint), but we will not give it here. The logarithmic loss of {(\log(1+|S|))^{O_k(1)}} was reduced to {\log(1+|S|)} by Mirek. In this paper we refine the bound further to

\displaystyle  \|T_{m;\Sigma_{\leq k}}\|_{B(\ell^p({\bf Z}))} \leq O(r \log(2+kr))^k \|T_m\|_{B(L^p({\bf R}))}. \ \ \ \ \ (3)

when {p = 2r} or {p = (2r)'} for some integer {r}. In particular there is no longer any logarithmic loss in the cardinality of the set {S}.

The proof of (3) follows a similar strategy as to previous proofs of Ionescu-Wainger type. By duality we may assume {p=2r}. We use the following standard sequence of steps:

  • (i) (Denominator orthogonality) First one splits {T_{m;\Sigma_{\leq k}} f} into various pieces depending on the denominator {Q} appearing in the element of {\Sigma_{\leq k}}, and exploits “superorthogonality” in {Q} to estimate the {\ell^p} norm by the {\ell^p} norm of an appropriate square function.
  • (ii) (Nonconcentration) One expands out the {p^{th}} power of the square function and estimates it by a “nonconcentrated” version in which various factors that arise in the expansion are “disjoint”.
  • (iii) (Numerator orthogonality) We now decompose based on the numerators {a} appearing in the relevant elements of {\Sigma_{\leq k}}, and exploit some residual orthogonality in this parameter to reduce to estimating a square-function type expression involving sums over various cosets {\alpha + \frac{1}{Q}{\bf Z}/{\bf Z}}.
  • (iv) (Marcinkiewicz-Zygmund) One uses the Marcinkiewicz-Zygmund theorem relating scalar and vector valued operator norms to eliminate the role of the multiplier {m}.
  • (v) (Rubio de Francia) Use a reverse square function estimate of Rubio de Francia type to conclude.

The main innovations are that of using the probabilistic decoupling method to remove some logarithmic losses in (i), and recent progress on the Erdos-Rado sunflower conjecture (as discussed in this recent post) to improve the bounds in (ii). For (i), the key point is that one can express a sum such as

\displaystyle  \sum_{A \in \binom{S}{k}} f_A,

where {\binom{S}{k}} is the set of {k}-element subsets of an index set {S}, and {f_A} are various complex numbers, as an average

\displaystyle  \sum_{A \in \binom{S}{k}} f_A = \frac{k^k}{k!} {\bf E} \sum_{s_1 \in {\bf S}_1,\dots,s_k \in {\bf S}_k} f_{\{s_1,\dots,s_k\}}

where {S = {\bf S}_1 \cup \dots \cup {\bf S}_k} is a random partition of {S} into {k} subclasses (chosen uniformly over all such partitions), basically because every {k}-element subset {A} of {S} has a probability exactly {\frac{k!}{k^k}} of being completely shattered by such a random partition. This “decouples” the index set {\binom{S}{k}} into a Cartesian product {{\bf S}_1 \times \dots \times {\bf S}_k} which is more convenient for application of the superorthogonality theory. For (ii), the point is to efficiently obtain estimates of the form

\displaystyle  (\sum_{A \in \binom{S}{k}} F_A)^r \lesssim_{k,r} \sum_{A_1,\dots,A_r \in \binom{S}{k} \hbox{ sunflower}} F_{A_1} \dots F_{A_r}

where {F_A} are various non-negative quantities, and a sunflower is a collection of sets {A_1,\dots,A_r} that consist of a common “core” {A_0} and disjoint “petals” {A_1 \backslash A_0,\dots,A_r \backslash A_0}. The other parts of the argument are relatively routine; see for instance this survey of Pierce for a discussion of them in the simple case {k=1}.

In this paper we interpret the Ionescu-Wainger multiplier theorem as being essentially a consequence of various quantitative versions of the Shannon sampling theorem. Recall that this theorem asserts that if a (Schwartz) function {f: {\bf R} \rightarrow {\bf C}} has its Fourier transform supported on {[-1/2,1/2]}, then {f} can be recovered uniquely from its restriction {f|_{\bf Z}: {\bf Z} \rightarrow {\bf C}}. In fact, as can be shown from a little bit of routine Fourier analysis, if we narrow the support of the Fourier transform slightly to {[-c,c]} for some {0 < c < 1/2}, then the restriction {f|_{\bf Z}} has the same {L^p} behaviour as the original function, in the sense that

\displaystyle  \| f|_{\bf Z} \|_{\ell^p({\bf Z})} \sim_{c,p} \|f\|_{L^p({\bf R})} \ \ \ \ \ (4)

for all {0 < p \leq \infty}; see Theorem 4.18 of this paper of myself with Krause and Mirek. This is consistent with the uncertainty principle, which suggests that such functions {f} should behave like a constant at scales {\sim 1/c}.

The quantitative sampling theorem (4) can be used to give an alternate proof of Proposition 1(i), basically thanks to the identity

\displaystyle  T_{m;0} (f|_{\bf Z}) = (T_m f)_{\bf Z}

whenever {f: {\bf R} \rightarrow {\bf C}} is Schwartz and has Fourier transform supported in {[-1/2,1/2]}, and {m} is also supported on {[-1/2,1/2]}; this identity can be easily verified from the Poisson summation formula. A variant of this argument also yields an alternate proof of Proposition 1(ii), where the role of {{\bf R}} is now played by {{\bf R} \times {\bf Z}/Q{\bf Z}}, and the standard embedding of {{\bf Z}} into {{\bf R}} is now replaced by the embedding {\iota_Q: n \mapsto (n, n \hbox{ mod } Q)} of {{\bf Z}} into {{\bf R} \times {\bf Z}/Q{\bf Z}}; the analogue of (4) is now

\displaystyle  \| f \circ \iota_Q \|_{\ell^p({\bf Z})} \sim_{c,p} \|f\|_{L^p({\bf R} \times {\bf Z}/Q{\bf Z})} \ \ \ \ \ (5)

whenever {f: {\bf R} \times {\bf Z}/Q{\bf Z} \rightarrow {\bf C}} is Schwartz and has Fourier transform {{\mathcal F}_{{\bf R} \times {\bf Z}/Q{\bf Z}} f\colon {\bf R} \times \frac{1}{Q}{\bf Z}/{\bf Z} \rightarrow {\bf C}} supported in {[-c/Q,c/Q] \times \frac{1}{Q}{\bf Z}/{\bf Z}}, and {{\bf Z}/Q{\bf Z}} is endowed with probability Haar measure.

The locally compact abelian groups {{\bf R}} and {{\bf R} \times {\bf Z}/Q{\bf Z}} can all be viewed as projections of the adelic integers {{\bf A}_{\bf Z} := {\bf R} \times \hat {\bf Z}} (the product of the reals and the profinite integers {\hat {\bf Z}}). By using the Ionescu-Wainger multiplier theorem, we are able to obtain an adelic version of the quantitative sampling estimate (5), namely

\displaystyle  \| f \circ \iota \|_{\ell^p({\bf Z})} \sim_{c,p} \|f\|_{L^p({\bf A}_{\bf Z})}

whenever {1 < p < \infty}, {f: {\bf A}_{\bf Z} \rightarrow {\bf C}} is Schwartz-Bruhat and has Fourier transform {{\mathcal F}_{{\bf A}_{\bf Z}} f: {\bf R} \times {\bf Q}/{\bf Z} \rightarrow {\bf C}} supported on {[-\varepsilon,\varepsilon] \times \Sigma_{\leq k}} for some sufficiently small {\varepsilon} (the precise bound on {\varepsilon} depends on {S, p, c} in a fashion not detailed here). This allows one obtain an “adelic” extension of the Ionescu-Wainger multiplier theorem, in which the {\ell^p({\bf Z})} operator norm of any discrete multiplier operator whose symbol is supported on major arcs can be shown to be comparable to the {L^p({\bf A}_{\bf Z})} operator norm of an adelic counterpart to that multiplier operator; in principle this reduces “major arc” harmonic analysis on the integers {{\bf Z}} to “low frequency” harmonic analysis on the adelic integers {{\bf A}_{\bf Z}}, which is a simpler setting in many ways (mostly because the set of major arcs (2) is now replaced with a product set {[-\varepsilon,\varepsilon] \times \Sigma_{\leq k}}).

Ben Krause, Mariusz Mirek, and I have uploaded to the arXiv our paper Pointwise ergodic theorems for non-conventional bilinear polynomial averages. This paper is a contribution to the decades-long program of extending the classical ergodic theorems to “non-conventional” ergodic averages. Here, the focus is on pointwise convergence theorems, and in particular looking for extensions of the pointwise ergodic theorem of Birkhoff:

Theorem 1 (Birkhoff ergodic theorem) Let {(X,\mu,T)} be a measure-preserving system (by which we mean {(X,\mu)} is a {\sigma}-finite measure space, and {T: X \rightarrow X} is invertible and measure-preserving), and let {f \in L^p(X)} for any {1 \leq p < \infty}. Then the averages {\frac{1}{N} \sum_{n=1}^N f(T^n x)} converge pointwise for {\mu}-almost every {x \in X}.

Pointwise ergodic theorems have an inherently harmonic analysis content to them, as they are closely tied to maximal inequalities. For instance, the Birkhoff ergodic theorem is closely tied to the Hardy-Littlewood maximal inequality.

The above theorem was generalized by Bourgain (conceding the endpoint {p=1}, where pointwise almost everywhere convergence is now known to fail) to polynomial averages:

Theorem 2 (Pointwise ergodic theorem for polynomial averages) Let {(X,\mu,T)} be a measure-preserving system, and let {f \in L^p(X)} for any {1 < p < \infty}. Let {P \in {\bf Z}[{\mathrm n}]} be a polynomial with integer coefficients. Then the averages {\frac{1}{N} \sum_{n=1}^N f(T^{P(n)} x)} converge pointwise for {\mu}-almost every {x \in X}.

For bilinear averages, we have a separate 1990 result of Bourgain (for {L^\infty} functions), extended to other {L^p} spaces by Lacey, and with an alternate proof given, by Demeter:

Theorem 3 (Pointwise ergodic theorem for two linear polynomials) Let {(X,\mu,T)} be a measure-preserving system with finite measure, and let {f \in L^{p_1}(X)}, {g \in L^{p_2}} for some {1 < p_1,p_2 \leq \infty} with {\frac{1}{p_1}+\frac{1}{p_2} < \frac{3}{2}}. Then for any integers {a,b}, the averages {\frac{1}{N} \sum_{n=1}^N f(T^{an} x) g(T^{bn} x)} converge pointwise almost everywhere.

It has been an open question for some time (see e.g., Problem 11 of this survey of Frantzikinakis) to extend this result to other bilinear ergodic averages. In our paper we are able to achieve this in the partially linear case:

Theorem 4 (Pointwise ergodic theorem for one linear and one nonlinear polynomial) Let {(X,\mu,T)} be a measure-preserving system, and let {f \in L^{p_1}(X)}, {g \in L^{p_2}} for some {1 < p_1,p_2 < \infty} with {\frac{1}{p_1}+\frac{1}{p_2} \leq 1}. Then for any polynomial {P \in {\bf Z}[{\mathrm n}]} of degree {d \geq 2}, the averages {\frac{1}{N} \sum_{n=1}^N f(T^{n} x) g(T^{P(n)} x)} converge pointwise almost everywhere.

We actually prove a bit more than this, namely a maximal function estimate and a variational estimate, together with some additional estimates that “break duality” by applying in certain ranges with {\frac{1}{p_1}+\frac{1}{p_2}>1}, but we will not discuss these extensions here. A good model case to keep in mind is when {p_1=p_2=2} and {P(n) = n^2} (which is the case we started with). We note that norm convergence for these averages was established much earlier by Furstenberg and Weiss (in the {d=2} case at least), and in fact norm convergence for arbitrary polynomial averages is now known thanks to the work of Host-Kra, Leibman, and Walsh.

Our proof of Theorem 4 is much closer in spirit to Theorem 2 than to Theorem 3. The property of the averages shared in common by Theorems 2, 4 is that they have “true complexity zero”, in the sense that they can only be only be large if the functions {f,g} involved are “major arc” or “profinite”, in that they behave periodically over very long intervals (or like a linear combination of such periodic functions). In contrast, the average in Theorem 3 has “true complexity one”, in the sense that they can also be large if {f,g} are “almost periodic” (a linear combination of eigenfunctions, or plane waves), and as such all proofs of the latter theorem have relied (either explicitly or implicitly) on some form of time-frequency analysis. In principle, the true complexity zero property reduces one to study the behaviour of averages on major arcs. However, until recently the available estimates to quantify this true complexity zero property were not strong enough to achieve a good reduction of this form, and even once one was in the major arc setting the bilinear averages in Theorem 4 were still quite complicated, exhibiting a mixture of both continuous and arithmetic aspects, both of which being genuinely bilinear in nature.

After applying standard reductions such as the Calderón transference principle, the key task is to establish a suitably “scale-invariant” maximal (or variational) inequality on the integer shift system (in which {X = {\bf Z}} with counting measure, and {T(n) = n-1}). A model problem is to establish the maximal inequality

\displaystyle  \| \sup_N |A_N(f,g)| \|_{\ell^1({\bf Z})} \lesssim \|f\|_{\ell^2({\bf Z})}\|g\|_{\ell^2({\bf Z})} \ \ \ \ \ (1)

where {N} ranges over powers of two and {A_N} is the bilinear operator

\displaystyle  A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^N f(x-n) g(x-n^2).

The single scale estimate

\displaystyle  \| A_N(f,g) \|_{\ell^1({\bf Z})} \lesssim \|f\|_{\ell^2({\bf Z})}\|g\|_{\ell^2({\bf Z})}

or equivalently (by duality)

\displaystyle  \frac{1}{N} \sum_{n=1}^N \sum_{x \in {\bf Z}} h(x) f(x-n) g(x-n^2) \lesssim \|f\|_{\ell^2({\bf Z})}\|g\|_{\ell^2({\bf Z})} \|h\|_{\ell^\infty({\bf Z})} \ \ \ \ \ (2)

is immediate from Hölder’s inequality; the difficulty is how to take the supremum over scales {N}.

The first step is to understand when the single-scale estimate (2) can come close to equality. A key example to keep in mind is when {f(x) = e(ax/q) F(x)}, {g(x) = e(bx/q) G(x)}, {h(x) = e(cx/q) H(x)} where {q=O(1)} is a small modulus, {a,b,c} are such that {a+b+c=0 \hbox{ mod } q}, {G} is a smooth cutoff to an interval {I} of length {O(N^2)}, and {F=H} is also supported on {I} and behaves like a constant on intervals of length {O(N)}. Then one can check that (barring some unusual cancellation) (2) is basically sharp for this example. A remarkable result of Peluse and Prendiville (generalised to arbitrary nonlinear polynomials {P} by Peluse) asserts, roughly speaking, that this example basically the only way in which (2) can be saturated, at least when {f,g,h} are supported on a common interval {I} of length {O(N^2)} and are normalised in {\ell^\infty} rather than {\ell^2}. (Strictly speaking, the above paper of Peluse and Prendiville only says something like this regarding the {f,h} factors; the corresponding statement for {g} was established in a subsequent paper of Peluse and Prendiville.) The argument requires tools from additive combinatorics such as the Gowers uniformity norms, and hinges in particular on the “degree lowering argument” of Peluse and Prendiville, which I discussed in this previous blog post. Crucially for our application, the estimates are very quantitative, with all bounds being polynomial in the ratio between the left and right hand sides of (2) (or more precisely, the {\ell^\infty}-normalized version of (2)).

For our applications we had to extend the {\ell^\infty} inverse theory of Peluse and Prendiville to an {\ell^2} theory. This turned out to require a certain amount of “sleight of hand”. Firstly, one can dualise the theorem of Peluse and Prendiville to show that the “dual function”

\displaystyle  A^*_N(h,g)(x) = \frac{1}{N} \sum_{n=1}^N h(x+n) g(x+n-n^2)

can be well approximated in {\ell^1} by a function that has Fourier support on “major arcs” if {g,h} enjoy {\ell^\infty} control. To get the required extension to {\ell^2} in the {f} aspect one has to improve the control on the error from {\ell^1} to {\ell^2}; this can be done by some interpolation theory combined with the useful Fourier multiplier theory of Ionescu and Wainger on major arcs. Then, by further interpolation using recent {\ell^p({\bf Z})} improving estimates of Han, Kovac, Lacey, Madrid, and Yang for linear averages such as {x \mapsto \frac{1}{N} \sum_{n=1}^N g(x+n-n^2)}, one can relax the {\ell^\infty} hypothesis on {g} to an {\ell^2} hypothesis, and then by undoing the duality one obtains a good inverse theorem for (2) for the function {f}; a modification of the arguments also gives something similar for {g}.

Using these inverse theorems (and the Ionescu-Wainger multiplier theory) one still has to understand the “major arc” portion of (1); a model case arises when {f,g} are supported near rational numbers {a/q} with {q \sim 2^l} for some moderately large {l}. The inverse theory gives good control (with an exponential decay in {l}) on individual scales {N}, and one can leverage this with a Rademacher-Menshov type argument (see e.g., this blog post) and some closer analysis of the bilinear Fourier symbol of {A_N} to eventually handle all “small” scales, with {N} ranging up to say {2^{2^u}} where {u = C 2^{\rho l}} for some small constant {\rho} and large constant {C}. For the “large” scales, it becomes feasible to place all the major arcs simultaneously under a single common denominator {Q}, and then a quantitative version of the Shannon sampling theorem allows one to transfer the problem from the integers {{\bf Z}} to the locally compact abelian group {{\bf R} \times {\bf Z}/Q{\bf Z}}. Actually it was conceptually clearer for us to work instead with the adelic integers {{\mathbf A}_{\bf Z} ={\bf R} \times \hat {\bf Z}}, which is the inverse limit of the {{\bf R} \times {\bf Z}/Q{\bf Z}}. Once one transfers to the adelic integers, the bilinear operators involved split up as tensor products of the “continuous” bilinear operator

\displaystyle  A_{N,{\bf R}}(f,g)(x) := \frac{1}{N} \int_0^N f(x-t) g(x-t^2)\ dt

on {{\bf R}}, and the “arithmetic” bilinear operator

\displaystyle  A_{\hat Z}(f,g)(x) := \int_{\hat {\bf Z}} f(x-y) g(x-y^2) d\mu_{\hat {\bf Z}}(y)

on the profinite integers {\hat {\bf Z}}, equipped with probability Haar measure {\mu_{\hat {\bf Z}}}. After a number of standard manipulations (interpolation, Fubini’s theorem, Hölder’s inequality, variational inequalities, etc.) the task of estimating this tensor product boils down to establishing an {L^q} improving estimate

\displaystyle  \| A_{\hat {\bf Z}}(f,g) \|_{L^q(\hat {\bf Z})} \lesssim \|f\|_{L^2(\hat {\bf Z})} \|g\|_{L^2(\hat {\bf Z})}

for some {q>2}. Splitting the profinite integers {\hat {\bf Z}} into the product of the {p}-adic integers {{\bf Z}_p}, it suffices to establish this claim for each {{\bf Z}_p} separately (so long as we keep the implied constant equal to {1} for sufficiently large {p}). This turns out to be possible using an arithmetic version of the Peluse-Prendiville inverse theorem as well as an arithmetic {L^q} improving estimate for linear averaging operators which ultimately arises from some estimates on the distribution of polynomials on the {p}-adic field {{\bf Q}_p}, which are a variant of some estimates of Kowalski and Wright.

The Riemann zeta function \zeta(s), defined for \hbox{Re}(s) > 1 by the formula

\displaystyle \zeta(s) := \sum_{n \in {\Bbb N}} \frac{1}{n^s} (1)

where {\Bbb N} = \{1,2,\ldots\} are the natural numbers, and extended meromorphically to other values of s by analytic continuation, obeys the remarkable functional equation

\displaystyle \Xi(s) = \Xi(1-s) (2)

where

\displaystyle \Xi(s) := \Gamma_\infty(s) \zeta(s) (3)

is the Riemann Xi function,

\displaystyle \Gamma_\infty(s) := \pi^{-s/2} \Gamma(s/2) (4)

is the Gamma factor at infinity, and the Gamma function \Gamma(s) is defined for \hbox{Re}(s) > 1 by

\displaystyle \Gamma(s) := \int_0^\infty e^{-t} t^s\ \frac{dt}{t} (5)

and extended meromorphically to other values of s by analytic continuation.

There are many proofs known of the functional equation (2).  One of them (dating back to Riemann himself) relies on the Poisson summation formula

\displaystyle \sum_{a \in {\Bbb Z}} f_\infty(a t_\infty) = \frac{1}{|t|_\infty} \sum_{a \in {\Bbb Z}} \hat f_\infty(a/t_\infty) (6)

for the reals k_\infty := {\Bbb R} and t \in k_\infty^*, where f is a Schwartz function, |t|_\infty := |t| is the usual Archimedean absolute value on k_\infty, and

\displaystyle \hat f_\infty(\xi_\infty) := \int_{k_\infty} e_\infty(-x_\infty \xi_\infty) f_\infty(x_\infty)\ dx_\infty (7)

is the Fourier transform on k_\infty, with e_\infty(x_\infty) := e^{2\pi i x_\infty} being the standard character e_\infty: k_\infty \to S^1 on k_\infty.  (The reason for this rather strange notation for the real line and its associated structures will be made clearer shortly.)  Applying this formula to the (Archimedean) Gaussian function

\displaystyle g_\infty(x_\infty) := e^{-\pi |x_\infty|^2}, (8)

which is its own (additive) Fourier transform, and then applying the multiplicative Fourier transform (i.e. the Mellin transform), one soon obtains (2).  (Riemann also had another proof of the functional equation relying primarily on contour integration, which I will not discuss here.)  One can “clean up” this proof a bit by replacing the Gaussian by a Dirac delta function, although one now has to work formally and “renormalise” by throwing away some infinite terms.  (One can use the theory of distributions to make this latter approach rigorous, but I will not discuss this here.)  Note how this proof combines the additive Fourier transform with the multiplicative Fourier transform.  [Continuing with this theme, the Gamma function (5) is an inner product between an additive character e^{-t} and a multiplicative character t^s, and the zeta function (1) can be viewed both additively, as a sum over n, or multiplicatively, as an Euler product.]

In the famous thesis of Tate, the above argument was reinterpreted using the language of the adele ring {\Bbb A}, with the Poisson summation formula (4) on k_\infty replaced by the Poisson summation formula

\displaystyle \sum_{a \in k} f(a t) = \sum_{a \in k} \hat f(t/a) (9)

on {\Bbb A}, where k = {\Bbb Q} is the rationals, t \in {\Bbb A}, and f is now a Schwartz-Bruhat function on {\Bbb A}.  Applying this formula to the adelic (or global) Gaussian function g(x) := g_\infty(x_\infty) \prod_p 1_{{\mathbb Z}_p}(x_p), which is its own Fourier transform, and then using the adelic Mellin transform, one again obtains (2).  Again, the proof can be cleaned up by replacing the Gaussian with a Dirac mass, at the cost of making the computations formal (or requiring the theory of distributions).

In this post I will write down both Riemann’s proof and Tate’s proof together (but omitting some technical details), to emphasise the fact that they are, in some sense, the same proof.  However, Tate’s proof gives a high-level clarity to the situation (in particular, explaining more adequately why the Gamma factor at infinity (4) fits seamlessly with the Riemann zeta function (1) to form the Xi function (2)), and allows one to generalise the functional equation relatively painlessly to other zeta-functions and L-functions, such as Dedekind zeta functions and Hecke L-functions.

[Note: the material here is very standard in modern algebraic number theory; the post here is partially for my own benefit, as most treatments of this topic in the literature tend to operate in far higher levels of generality than I would prefer.]

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