On Superorthogonality

Abstract

In this survey, we explore how superorthogonality amongst functions in a sequence $f_1,f_2,f_3,\ldots$ results in direct or converse inequalities for an associated square function. We distinguish between three main types of superorthogonality, which we demonstrate arise in a wide array of settings in harmonic analysis and number theory. This perspective gives clean proofs of central results, and unifies topics including Khintchine’s inequality, Walsh–Paley series, discrete operators, decoupling, counting solutions to systems of Diophantine equations, multicorrelation of trace functions, and the Burgess bound for short character sums.

Appendices

Appendix A: Further Remarks on Walsh–Paley Series

We deferred a few details on the direct and converse inequalities in the setting of Walsh–Paley series in Sect. 4. Here, we first remark on the limiting argument to obtain (4.4) for $p=2r$ from the truncated version (4.21). Second, we remark on deducing the cases for $1<p<\infty$ from the cases with p an even integer; this illustrates a further application of Khintchine’s inequality. Third, we show how to deduce the operator bound (4.5) from the dyadic direct and converse inequalities (4.4).

1.1 A.1. Limiting Arguments for Direct and Converse Inequalities

Fix $p=2r$. In the main text we showed that uniformly in N,

$$\begin{aligned} \left\| \sum _{n=0}^N f_n \right\| _{L^p} \le c_p \left\| \left( \sum _{n=0}^N f_n^2\right) ^{1/2} \right\| _{L^p} \le c_p \left\| \left( \sum _{n=0}^\infty f_n^2\right) ^{1/2} \right\| _{L^p}. \end{aligned}$$

The same method of proof used to obtain this shows that for any $N_1< N_2$,

$$\begin{aligned} \Vert S_{2^{N_2}}f - S_{2^{N_1}} f \Vert _{L^p}\le c_p \left\| \left( \sum _{n=N_1+1}^{N_2} f_n^2\right) ^{1/2} \right\| _{L^p}. \end{aligned}$$

If f is such that the right-hand side of the direct inequality converges, then this tail must vanish as $N_1, N_2 \rightarrow \infty$, so that as $N \rightarrow \infty$, $S_{2^N}f$ converges in $L^p$ norm to some function, say F, which satisfies $\Vert F\Vert _{L^p} \le c_p\left\| \left( \sum _{n=0}^\infty f_n^2\right) ^{1/2}\right\| _{L^p} .$ By the Dominated Convergence Theorem, for each m

$$\begin{aligned} c_m(F) = \int _0^1 F(\theta ) w_m(\theta ) \mathrm{d}\theta = \int _0^1 f(\theta )w_m(\theta ) \mathrm{d}\theta = c_m(f), \end{aligned}$$

and since $\{w_m\}$ is a complete orthonormal system on [0, 1], we conclude $F=f$, verifying the direct inequality. For the converse inequality, we apply the maximal bound (4.20) to see that $\left\| \left( \sum _{n=0}^N f_n^2\right) ^{1/2} \right\| _{L^p} \le c_p' \left\| \sum _{n=0}^N f_n \right\| _{L^p} \ll _p \Vert f\Vert _{L^p}$ uniformly in N, which suffices.

1.2 A.2 Linearization

We have verified the direct and converse inequalities (4.4) in $L^p$ for each even integer $p \ge 2$. To conclude the results for all $1< p <\infty$, we recall Paley’s arguments (now standard), in which the Rademacher functions again make an appearance, via Khintchine’s inequality.

One would like to interpolate either the direct inequality (or the converse inequality, respectively), but one must first linearize. For any fixed $1<p<\infty$, the truth for all $f \in L^p$ of the direct and converse inequalities

$$\begin{aligned} \left\| \left( \sum _{n=0}^\infty f_n^2\right) ^{1/2} \right\| _{L^p} \ll _p \Vert f\Vert _{L^p} \ll _p \left\| \left( \sum _{n=0}^\infty f_n^2\right) ^{1/2} \right\| _{L^p} \end{aligned}$$

(A.1)

is equivalent to the truth of the statement that

$$\begin{aligned} \Vert f^*\Vert _{L^p}\ll _p \Vert f\Vert _{L^p} \ll _p \Vert f^*\Vert _{L^p} \end{aligned}$$

(A.2)

holds for all $f \in L^p$, uniformly for all choices of $\varepsilon _n \in \{ \pm 1\}$, where

$$\begin{aligned} f^*(t) = \sum _{n=0}^\infty \varepsilon _n f_n(t). \end{aligned}$$

The advantage of (A.2) is that the expressions in this inequality are linear, and thus well-suited to interpolation.

Let us verify the equivalence. If (A.2) holds, to deduce (A.1), we use the Rademacher functions. Given f and its associated sequence $\{f_n\}$ we define an auxiliary function $F(t,\theta ) = \sum _{n=0}^\infty r_n(\theta ) f_n(t)$ for each $\theta \in [0,1]$. By assumption of (A.2), for each fixed $\theta$,

$$\begin{aligned} \int _0^1 | F(t,\theta )|^p \mathrm{d}t \ll _p \int _0^1 |f(t)|^p \mathrm{d}t \ll _p \int _0^1 | F(t,\theta )|^p \mathrm{d}t. \end{aligned}$$

We integrate this over $\theta \in [0,1]$ to conclude by Fubini’s theorem that

$$\begin{aligned} \int _0^1 \int _0^1 \left| \sum _{n=0}^\infty r_n(\theta ) f_n(t)\right| ^p \mathrm{d}\theta \mathrm{d}t \ll _p \int _0^1 |f(t)|^p \mathrm{d}t \ll _p \int _0^1 \int _0^1 \left| \sum _{n=0}^\infty r_n(\theta ) f_n(t)\right| ^p\mathrm{d}\theta \mathrm{d}t. \end{aligned}$$

Now for each fixed t we apply Khintchine’s inequality (2.5), and this proves that (A.1) holds, as desired.

The converse is more elementary. Given $f \in L^p$, and any choice of $\{\varepsilon _n\}$, $f^*$ is the function with associated expansion $\sum _{n=0}^\infty g_n$ with $g_n=\varepsilon _n f_n$, so that applying the direct inequality followed by the converse inequality assumed in (A.1) shows that

$$\begin{aligned} \Vert f^* \Vert _{L^p} \ll _p \left\| \left( \sum _n g_n^2\right) ^{1/2} \right\| _{L^p} =\left\| \left( \sum _n f_n^2\right) ^{1/2} \right\| _{L^p} \ll _p \Vert f\Vert _{L^p}. \end{aligned}$$

One obtains $\Vert f\Vert _{L^p} \ll _p \Vert f^*\Vert _{L^p}$ in an analogous fashion.

1.3 A.3 Remarks for $2 \le p < \infty$

We know that (A.1) and hence (A.2) holds for each $p=2r$ with $r \ge 1$ an integer. We fix a sequence $\{ \varepsilon _n\}_n$ with $\varepsilon _n \in \{\pm 1\}$ and consider a truncation $(S_{2^N}f)^* (t) = \sum _{0 \le n \le N} \varepsilon _n f_n(t)$. Then applying the left-hand side of (A.2), for every even integer $p \ge 2$,

$$\begin{aligned} \Vert (S_{2^N}f)^* \Vert _{L^p} \ll _p \Vert S_{2^N}f\Vert _{L^p} \ll _p\Vert f\Vert _{L^p}, \end{aligned}$$

in which the last inequality holds uniformly in N, by the maximal theorem in (4.20). By Riesz–Thorin interpolation between $p=2$ and any even integer, we conclude that this inequality holds for all $2 \le p < \infty$. For a fixed $p \ge 2$, we can then deduce that $(S_{2^N}f)^*$ converges in $L^p$ norm to a limit function, say $F^*$. By the Dominated Convergence Theorem, the coefficients $c_m(F^*)$ agree with those of $f^*$, and since the Walsh functions form a complete system, we learn that $F^*=f^*$. We conclude that $\Vert f^*\Vert _{L^p} \ll _p \Vert f\Vert _{L^p}$, obtaining the left-hand inequality of (A.2) for each $2\le p < \infty$. For the other inequality, we simply observe that given f and a fixed sequence $\{\varepsilon _n\}$, then $(f^*)^*=f$, so the right-hand inequality of (A.2) follows.

1.4 A.4 Remarks for $1< p \le 2$

One again uses the linearized inequalities (A.2) in order to apply duality. Fix $1<p \le 2$, and fix a sequence of $\varepsilon _n \in \{\pm 1\}$, and accordingly define $f_N^* = \sum _{0 \le n \le N} \varepsilon _n f_n$. By duality, to show that $\Vert f_N^*\Vert _{L^p} \ll _p \Vert f\Vert _{L^p}$ it suffices to show that for all $g \in L^{p'}$ with $1/p + 1/p'=1$, $\Vert f_N^* g\Vert _{L^1} \ll _p \Vert g\Vert _{L^{p'}} \Vert f\Vert _{L^p}.$ Precisely,

$$\begin{aligned} \left\| \left( \sum _{n=0}^{N} \varepsilon _n f_n\right) g \right\| _{L^1} = \left\| \left( \sum _{n=0}^{N} \varepsilon _n g_n\right) f \right\| _{L^1} \le \left\| \sum _{n=0}^{N} \varepsilon _n g_n\right\| _{L^{p'}} \left\| {f}\right\| _{L^p}, \end{aligned}$$

with the last inequality due to Hölder’s inequality. We apply the known case for $p' \ge 2$, so that $\left\| \sum _{n=0}^{N} \varepsilon _n g_n\right\| _{L^{p'}}\ll _p \Vert g\Vert _{L^{p'}}$, uniformly in the choice of signs $\{ \varepsilon _n\}$. We conclude that $\Vert f_N^*\Vert _{L^p} \ll _p \Vert f\Vert _{L^p}$ uniformly in N, and uniformly in the choice of $\{\varepsilon _n\}$. Thus we may argue as before that $f_N^*$ converges in $L^p$ norm to a function, which we may check is indeed $f^* = \sum \varepsilon _n f_n$, and this verifies that $\Vert f^*\Vert _{L^p} \ll _p \Vert f\Vert _{L^p}$ holds. For the other inequality, we again note that for each fixed choice of signs, $(f^*)^*=f$, and thus we obtain $\Vert f\Vert _{L^p} \ll _p \Vert f^*\Vert _{L^p}$, concluding the proof.

1.5 A.5 Combining the Direct and Converse Inequalities

Fix $1<p<\infty$ and $n \ge 1$. To combine the direct and converse inequalities for the dyadic differences $f_n = S_{2^{n}}f - S_{2^{n-1}}f$ in order to bound $S_n f$ on $L^p$, we must be able to express the partial sum $S_nf$ in terms of dyadic differences. Paley employs an identity of the following flavor. Write the binary expansion $n=2^{n_1} + \cdots + 2^{n_s}$ with $n_1> \cdots > n_s$. We claim

$$\begin{aligned} w_n(t) w_n(\theta ) \sum _{m=0}^{n-1} w_m(t)w_m(\theta )= & {} \sum _{m \in [2^{n_1}, 2^{n_1+1})} w_m(t)w_m(\theta ) \nonumber \\+ & {} \cdots +\sum _{m \in [2^{n_s}, 2^{n_s+1})} w_m(t)w_m(\theta ). \end{aligned}$$

(A.3)

Once we have verified this, the deduction is simple. Recall

$$\begin{aligned} S_n f (t) = \sum _{m=0}^{n-1} c_m(f) w_m(t) = \int _0^1 f(\theta ) \sum _{m=0}^{n-1} w_m(\theta ) w_m(t) \mathrm{d}\theta . \end{aligned}$$

To introduce the extraneous factor $w_n$ which is critical to the identity (A.3), given any $f \in L^p[0,1]$ we define the function $g(\theta ) = f(\theta ) w_n(\theta )$ with identical $L^p$ norm; we will also use the notation $g_m = S_{2^{m}} g - S_{2^{m-1}} g.$ Then using $w_n(\theta )^2 \equiv 1$ followed by (A.3),

$$\begin{aligned} w_n(t) (S_n f)(t) = \int _0^1 g(\theta ) w_n(\theta ) w_n(t) \sum _{m=0}^{n-1} w_m(\theta ) w_m(t) \mathrm{d}\theta = g_{n_1+1}(t) + \cdots + g_{n_s+1}(t). \end{aligned}$$

Now applying first the direct inequality and then the converse inequality for the functions $\{g_n\}$ we obtain the desired result:

$$\begin{aligned} \Vert S_nf \Vert _{L^p}= & {} \Vert \sum _{j=1}^s g_{n_j+1} \Vert _{L^p} \ll _p \left\| \left( \sum _{j=1}^s g_{n_j+1}^2\right) ^{1/2} \right\| _{L^p}\\\le & {} \left\| \left( \sum _{n=0}^\infty g_n^2\right) ^{1/2} \right\| _{L^p} \ll _p \Vert g \Vert _{L^p} = \Vert f \Vert _{L^p}. \end{aligned}$$

To verify (A.3), it suffices to observe an equivalent identity about sets of numbers written in binary (also expressible in terms of properties of the Walsh group or “dyadic group,” see [23, §2] or [3]). Precisely, fix n and $m \le n$ and suppose $n=2^{n_1} + \cdots + 2^{n_s}$ (with $n_1> \cdots > n_s$) and $m=2^{m_1} + \cdots + 2^{m_r}$ (with $m_1> \cdots > m_r$), and let the $(n_1+1)$-digit representation of n and m in binary be ${\underline{n}}, {\underline{m}}$, respectively. Then $w_n w_m = w_u$ where ${\underline{u}} = {\underline{n}} \oplus {\underline{m}}$; here $\oplus$ denotes exclusive-or summation. (Since the square of any Rademacher function is identically one, if any exponent occurs in both the binary expansion of n and of m, then it does not appear as an exponent in the binary expansion of u for the function $w_u$ such that $w_u = w_nw_m$.)

Consequently, (A.3) is equivalent to the following identity on sets of distinct binary numbers:

$$\begin{aligned} \{ {\underline{n}} \oplus {\underline{m}} : 0 \le m< n \} = \bigsqcup _{j=1}^s \{ {\underline{m}} : 2^{n_j} \le m < 2^{n_j+1}\}. \end{aligned}$$

We can first verify that for $j=1$, $\{ {\underline{n}} \oplus {\underline{m}} : 0 \le m< 2^{n_1} \} = \{ {\underline{m}} : 2^{n_1} \le m < 2^{n_1+1}\}.$ This is because the map acting on $0 \le m < 2^{n_1}$ by $m \mapsto {\underline{n}} \oplus {\underline{m}}$ is injective and maps into $\{ {\underline{m}} : 2^{n_1} \le m < 2^{n_1+1}\}$; since the cardinalities match, it is a bijection. Similarly, one can see that for each $2 \le j \le s$,

$$\begin{aligned} \{ {\underline{n}} \oplus {\underline{m}} : 2^{n_1} + \cdots + 2^{n_{j-1}} \le m< 2^{n_1} + \cdots + 2^{n_{j-1}} + 2^{n_j} \} = \{ {\underline{m}} : 2^{n_j} \le m < 2^{n_j+1}\}, \end{aligned}$$

and the claim holds.

In Remark 4.3, we claimed that while the functions $\{w_n\}$ are orthogonal, they do not themselves possess superorthogonality properties for 2r-tuples with $r\ge 2$. This referred to the fact that for any $r \ge 2$, we can pick 2r functions $w_n$ with 2r distinct values of n (so the tuple $(n_1,\ldots , n_{2r})$ satisfies the hypothesis of Type I or Type II or Type III) such that $\int w_{n_1} \cdots w_{n_{2r}} = 1$. Using the notation introduced above, this follows from the fact that we can choose 2r pairwise distinct integers $n_1, \ldots , n_{2r}$ such that when written in binary, ${\underline{n}}_1 \oplus \cdots \oplus {\underline{n}}_{2r}=0$.

Appendix B: The Source of Quasi-superorthogonality for Trace Functions

Appendix by Emmanuel Kowalski ¹

This short note will attempt to explain the source of the quasi-superorthogonality of trace functions that appears in Sect. 7, and in particular it will highlight that it arises from “exact” superorthogonality (of the corresponding type) for other functions, combined with Deligne’s very deep work on the Riemann Hypothesis over finite fields. We then explain briefly the source of the exact superorthogonality in the type of examples considered in the survey [25] of Fouvry, Kowalski and Michel.

Remark

The presentation is not fully rigorous, since we did not want to obscure the key conceptual point with technical aspects, such as the need to work with continuous $\ell$-adic representations, etc.

Let q be a prime number. The key data is a certain compact topological group $\varPi _q$ associated to q, with a normal subgroup $\varPi _q^g$ (both are algebraic variants of the classical fundamental group of topology, but mainly viewed as classifying coverings of the space, instead of groups of homotopy classes of loops). Moreover, for every $x\in {\mathbb {Z}}/q{\mathbb {Z}}$, there exists a conjugacy class $\theta _{q}(x)$ in $\varPi _q$ (called the Frobenius conjugacy class at x), and $\varPi _q^g$ is big enough that it and a single Frobenius conjugacy class generate $\varPi _q$ topologically.

A trace function F modulo q always has the following form: there exists a finite-dimensional vector space V on which $\varPi _q$ acts linearly (i.e., a finite-dimensional representation of the group) in such a way that

$$\begin{aligned} F(x)={{\,\mathrm{tr}\,}}(\theta _{q}(x)\mid V), \end{aligned}$$

(B.1)

the trace of the endomorphism of V associated to the Frobenius conjugacy class at x. This is well-defined, since the trace is invariant under conjugation.

We view the action as a homomorphism $\varrho :\varPi _q\rightarrow \mathrm {GL}(V)$. Then the formula (B.1) shows that a trace function is the restriction of the character of a representation to a certain subset of conjugacy classes of that group.²

The Grothendieck–Lefschetz trace formula combined with Deligne’s Riemann Hypothesis can then be shown to imply (for suitable trace functions) the statement that

$$\begin{aligned} \sum _{x\in {\mathbb {Z}}/q{\mathbb {Z}}} F(x)=\Bigl (\int _{\varPi _q^g}{{\,\mathrm{tr}\,}}(\varrho (y))\mathrm{d}y\Bigr )cq+O(\sqrt{q}), \end{aligned}$$

(B.2)

for some complex number c with $|c|\le 1$, where the integral is with respect to the probability Haar measure on the compact group $\varPi _q^g$ and the implied constant in the $O(\cdot )$ symbol depends only on “local” invariants of $\varrho$ which are usually easy to bound.

Remark

In many cases of interest, one deals with an action of $\varPi _q$ which has the property that $\varrho (\varPi _q^g)=\varrho (\varPi _q)$. Then (B.2) holds with $c=1$, and thus it indicates that the discrete sum of the trace of $\varrho$ over the finitely many Frobenius classes $\theta _q(x)$ is close to the integral over the whole group [note that $\varrho (\theta _q(x))\in \varPi ^g_q$ because of the assumption on $\varrho$]. However, the formula (B.2) holds in general in the stated form.

We can now explain how this, together with algebraic properties of certain compact Lie groups, leads to quasi-superorthogonality.

Suppose we have finitely many trace functions $F_1$, ..., $F_{2r}$, each associated to a representation $\varrho _i$ (on the space $V_i$), satisfying suitable conditions. We want to understand the sum

$$\begin{aligned} \sum _{x\in {\mathbb {Z}}/q{\mathbb {Z}}} F_1(x)\overline{F_2(x)}\cdots F_{2r-1}(x)\overline{F_{2r}(x)}. \end{aligned}$$

Part of the unspecified properties required of $\varrho _i$ imply that the contragradient or dual representation $D(\varrho _i)$ of $\varrho _i$ satisfies

$$\begin{aligned} {{\,\mathrm{tr}\,}}(D(\varrho _i)(y))=\overline{{{\,\mathrm{tr}\,}}(\varrho _i(y))}. \end{aligned}$$

So, according to (B.2), applied to the representation

$$\begin{aligned} \varrho =\varrho _1\otimes D(\varrho _2)\otimes \cdots \otimes \varrho _{2r-1}\otimes D(\varrho _{2r}), \end{aligned}$$

we get

$$\begin{aligned}&\sum _{x\in {\mathbb {Z}}/q{\mathbb {Z}}} F_1(x)\overline{F_2(x)}\cdots F_{2r-1}(x)\overline{F_{2r}(x)} \\&\quad = \Bigl (\int _{\varPi _q^g}{{\,\mathrm{tr}\,}}(\varrho _1(y)) \overline{{{\,\mathrm{tr}\,}}(\varrho _2(y))}\cdots {{\,\mathrm{tr}\,}}(\varrho _{2r-1}(y)) \overline{{{\,\mathrm{tr}\,}}(\varrho _{2r}(y))} dy\Bigr )c'q+O(\sqrt{q}), \end{aligned}$$

for some complex number $c'$ with $|c'|\le 1$.

Thus, we will obtain quasi-superorthogonality, of any type, for the trace functions, provided the characters ${{\,\mathrm{tr}\,}}(\varrho _i)$ of the $\varrho _i$ (restricted to the subgroup $\varPi _q^g$) satisfy exact superorthogonality of the same type.

We present now one source of such superorthogonality that lies behind many examples (but not all—for Dirichlet characters, such as in the inequality (7.9), the mechanism is a bit different).

In fact, at this point, we can replace $\varPi ^g_q$ by any fixed compact group G, with the $\varrho _i$ being unitary (continuous) finite-dimensional representations of G.

According to the character theory of compact groups the integral

$$\begin{aligned} \int _{G}{{\,\mathrm{tr}\,}}(\varrho _1(y)) \overline{{{\,\mathrm{tr}\,}}(\varrho _2(y))}\cdots {{\,\mathrm{tr}\,}}(\varrho _{2r-1}(y)) \overline{{{\,\mathrm{tr}\,}}(\varrho _{2r}(y))} \mathrm{d}y \end{aligned}$$

(B.3)

is equal to the dimension of the space of invariant vectors in the tensor product representation $\varrho$. Now suppose that each $V_i$ has dimension at least 2 and that the image of each $\varrho _i$, which is a subgroup of the unitary group of the space $V_i$, happens to be the special unitary group ${{\,\mathrm{SU}\,}}(V_i)$. Consider the map

$$\begin{aligned} y\mapsto (\varrho _1(y),\ldots ,\varrho _{2r}(y)) \end{aligned}$$

from G to

$$\begin{aligned} {{\,\mathrm{SU}\,}}(V_1)\times \cdots \times {{\,\mathrm{SU}\,}}(V_{2r}). \end{aligned}$$

Let H be its image. It is again a compact group, and it has the property that the projection of H to each factor ${{\,\mathrm{SU}\,}}(V_i)$ is surjective. Now a special case of what Katz [44, §1.8, Prop. 1.8.2] has called the Goursat–Kolchin–Ribet property is that such a subgroup H is equal to the product

$$\begin{aligned} {{\,\mathrm{SU}\,}}(V_1)\times \cdots \times {{\,\mathrm{SU}\,}}(V_{2r}), \end{aligned}$$

unless at least two of the representations are equivalent, in which case at least two of the characters ${{\,\mathrm{tr}\,}}(\varrho _i)$ are the same functions. (To see that this may be the case, consider the special case where all $V_i$ have different dimensions; then the groups ${{\,\mathrm{SU}\,}}(V_i)$ are pairwise non-isomorphic “almost” simple groups, and the projection assumption implies that the group H has to contain all of them as “Jordan–Hölder factors”, which is only possible if H is the full product.)

Thus, if no two of the characters are equal, then we have a splitting of the integral

$$\begin{aligned}&\int _{G}{{\,\mathrm{tr}\,}}(\varrho _1(y)) \overline{{{\,\mathrm{tr}\,}}(\varrho _2(y))}\cdots {{\,\mathrm{tr}\,}}(\varrho _{2r-1}(y)) \overline{{{\,\mathrm{tr}\,}}(\varrho _{2r}(y))} \mathrm{d}y \\&\quad = \int _H {{\,\mathrm{tr}\,}}(y_1,y_2^*,\ldots ,y_{2r-1},y_{2r}^*)\mathrm{d}y_1\cdots \mathrm{d}y_{2r} \\&\quad = \Bigl (\int _{{{\,\mathrm{SU}\,}}(V_1)}{{\,\mathrm{tr}\,}}(y_1)\mathrm{d}y_1\Bigr )\cdots \Bigl (\int _{{{\,\mathrm{SU}\,}}(V_{2r})}\overline{{{\,\mathrm{tr}\,}}(y_{2r})}\mathrm{d}y_{2r}\Bigr ), \end{aligned}$$

which vanishes. In other words, in these conditions, we obtain superorthogonality of Type II, and in fact really in the same way suggested at the beginning of the paper, i.e., from independent random variables, these being the different characters $y\mapsto {{\,\mathrm{tr}\,}}(\varrho _i(y))$.

One can be more precise about conditions on the representations $\varrho _i$ that lead to vanishing of the integral (B.3), but we hope that this sketch has given some idea of how this may arise.