Approximation of integration over finite groups, difference sets and association schemes

Abstract

Let G be a finite group and $f:G \rightarrow {\mathbb {C}}$ be a function. For a non-empty finite subset $Y\subset G$, let $I_Y(f)$ denote the average of f over Y. Then, $I_G(f)$ is the average of f over G. Using the decomposition of f into irreducible components of ${\mathbb {C}}^G$ as a representation of $G\times G$, we define non-negative real numbers V(f) and D(Y), each depending only on f, Y, respectively, such that an inequality of the form $|I_G(f)-I_Y(f)|\le V(f)\cdot D(Y)$ holds. We give a lower bound of D(Y) depending only on $\#Y$ and $\#G$. We show that the lower bound is achieved if and only if $\#\{(x,y)\in Y^2 \mid x^{-1}y \in [a]\}/\#[a]$ is independent of the choice of the conjugacy class $[a]\subset G$ for $a \ne 1$. We call such a $Y\subset G$ as a pre-difference set in G, since the condition is satisfied if Y is a difference set. If G is abelian, the condition is equivalent to that Y is a difference set. We found a non-trivial pre-difference set in the dihedral group of order 16, where no non-trivial difference set exists. The pre-difference sets in non-abelian groups of order 16 are classified. A generalization to commutative association schemes is also given.

Appendix A. quasi-Monte Carlo integration and characters

Here we explain a typical example of QMC integration. Put $X:=({\mathbb {R}}/{\mathbb {Z}})^s$. We consider a periodic real valued function of s-variables $f: X \rightarrow {\mathbb {R}}$. Let $\alpha$ be a positive integer, and ${\textbf{r}}=(r_1,\ldots ,r_s)$. We write ${\textbf{r}}\le \alpha$ if $r_i\le \alpha$ holds for each $1\le i \le s$. Let ${\textbf{x}}=(x_1,\ldots ,x_s) \in ({\mathbb {R}}/{\mathbb {Z}})^s$. For $f({\textbf{x}})$, we define

$$\begin{aligned} D^{\textbf{r}}(f):= \frac{\partial ^{r_1+\cdots +r_s}}{\partial x_1^{r_1}\cdots \partial x_s^{r_s}}(f) \end{aligned}$$

if it exists. We assume that $D^{\textbf{r}}(f)$ exists and is continuous for ${\textbf{r}}\le \alpha$. We define a norm

$$\begin{aligned} ||f||_\alpha :=\sum _{0\le {\textbf{r}}\le \alpha } ||D^{\textbf{r}}(f)||_{L^\infty }. \end{aligned}$$

The set ${\widehat{X}}$ of characters of X is

$$\begin{aligned} {\widehat{X}}= \{E_{\textbf{h}}({\textbf{x}}):=\exp (2\pi i {\textbf{h}}\cdot {\textbf{x}}) \ | \ {\textbf{h}}= (h_1,\ldots ,h_s) \in {\mathbb {Z}}^s \}. \end{aligned}$$

Let

$$\begin{aligned} f({\textbf{x}})=\sum _{{\textbf{h}}\in {\mathbb {Z}}^s}\hat{f}({\textbf{h}})E_{\textbf{h}}({\textbf{x}}) \end{aligned}$$

be the Fourier-expansion of f. Note that $\hat{f}(0)=I(f):=\int _{X} f({\textbf{x}})$. Let P be a finite subset in X. The QMC integration of f by P is the the average

$$\begin{aligned} I(f;P):=\frac{1}{\#P}\sum _{{\textbf{x}}\in P}f({\textbf{x}}), \end{aligned}$$

and the integration error is

$$\begin{aligned} \textrm{Err}(f;P)= & {} |I(f)-I(f;P)| =\left| \hat{f}(0)-\sum _{{\textbf{h}}\in {\mathbb {Z}}^s} \hat{f}({\textbf{h}})I(E_{\textbf{h}};P)\right| \nonumber \\\le & {} \sum _{{\textbf{h}}\in {\mathbb {Z}}^s-\{0\}} \left| \hat{f}({\textbf{h}})\right| \cdot \left| I(E_{\textbf{h}};P)\right| . \end{aligned}$$

(1)

Let

$$\begin{aligned} \rho ({\textbf{h}}):=\max \{1,|h_1|\}\times \cdots \times \max \{1,|h_s|\}. \end{aligned}$$

It is not difficult to show that the inequalities on the Fourier coefficients

$$\begin{aligned} |\hat{f}({\textbf{h}})|\le C_{s,\alpha } ||f||_\alpha \rho ({\textbf{h}})^{-\alpha } \end{aligned}$$

(2)

hold for a constant $C_{s,\alpha }$ depending only on $s,\alpha$ (cf. [6, §2.2][18, §.5.2.2]), and we have a Koksma-Hlawka type inequality on the error bound:

$$\begin{aligned} \textrm{Err}(f;P)\le C_{s,\alpha }||f||_\alpha \times \sum _{{\textbf{h}}\in {\mathbb {Z}}^s-\{0\}}\left| \rho ({\textbf{h}})^{-\alpha } I(E_{\textbf{h}};P)\right| . \end{aligned}$$

(3)

Thus, we want a point set P that makes $\sum _{{\textbf{h}}\in {\mathbb {Z}}^s-\{0\}}\left| \rho ({\textbf{h}})^{-\alpha } I(E_{\textbf{h}};P)\right|$ small. This is attained if $|I(E_{\textbf{h}};P)|$ is small (or ideally 0) for ${\textbf{h}}$ with small $\rho ({\textbf{h}})$.

From now on, we assume that $P\subset X$ is a finite cyclic subgroup of order N. Such a point set is called a rank-1 lattice and well-studied; see a nice introduction [17]. Let $P^\perp \subset {\mathbb {Z}}^s \cong {\widehat{X}}$ be the subgroup defined by

$$\begin{aligned} P^\perp :=\{{\textbf{h}}\in {\mathbb {Z}}^s \ | \ E_{\textbf{h}}({\textbf{x}})=1 \text{ for } \text{ all }\ {\textbf{x}}\in P\}. \end{aligned}$$

It is easy to show that $I(E_{\textbf{h}};P)=0$ if ${\textbf{h}}\notin P^\perp$, and $I(E_{\textbf{h}};P)=1$ if ${\textbf{h}}\in P^\perp$. Thus, we obtain the error-bound

$$\begin{aligned} \textrm{Err}(f;P)\le C_{s,\alpha }||f||_\alpha \times \sum _{{\textbf{h}}\in P^{\perp }-\{0\}}\left| \rho ({\textbf{h}})^{-\alpha } \right| . \end{aligned}$$

It is known that for any N there are P such that the summation in the right hand side is bounded by $C_{s,\alpha }'N^{-\alpha }(\log N)^{s\alpha }$ [14, Chapter 5], which implies that when N is increased, the error-bound decreases with order $O(N^{-\alpha }(\log N)^{s\alpha })$.

When compared with our study, the above error analysis (1) is essentially obtained from the left inequality in Corollary 2 (if we neglect that we treat only finite groups), where $\rho \in \widehat{G}$ corresponds to $E_{\textbf{h}}\in {\widehat{X}}$, $f_\rho$ corresponds to $\hat{f}({\textbf{h}})$, Y corresponds to P, and $\partial _\rho (Y)=||{\mathcal {I}}_Y^\rho || =|\langle E_\rho , {\mathcal {I}}_Y\rangle |= |I(E_\rho ;Y)|$ corresponds to $|I(E_{\textbf{h}};P)|$. A big difference is that as in (2), $\hat{f}({\textbf{h}})$ (the ${\textbf{h}}$-component of f) decays when $\rho ({\textbf{h}})$ gets large, and to make the error bound smaller, it is better to choose P such that $|I(E_{\textbf{h}};P)|=0$ holds for ${\textbf{h}}$ with small $\rho ({\textbf{h}})$ since $I(E_{\textbf{h}};P)$ has a large “weight” in the bound (3) (this condition is close to the idea of Delsarte’s “designs” in [3, §3.4]), while in our study, as in Theorem 3, we have no reasonable “weight” on characters (i.e. degree of importance of each character), and we need to treat them with equal importance, which leads to the notion of pre-difference sets. At present, we think that our results treating general groups and association schemes are rather wide and abstract (say, compared with $({\mathbb {R}}/{\mathbb {Z}})^s$) and that a practical application to QMC is a future work.

We here briefly explain the notion of digital nets in the theory of QMC [7, 14], which are widely used and closely related with character theory. The unit hypercube $[0,1]^s$ is approximated by $({\mathbb {F}}_2^n)^s$ via two-adic expansion upto n digits. A digital net P is a subgroup of $({\mathbb {F}}_2^n)^s$, identified as a subset of the hypercube. An important figure-of-merit of P is its t-value ([14, Chapter 4]), which is obtained from the minimum weight of $P^\perp$ with respect to Niederreiter-Rosenbloom-Tsfasman(NRT) metric [15], which is a generalization of the Hamming weight. P has a good (large) t-value if and only if $I(E_\rho ;P)=0$ holds for every $\rho \in \widehat{({\mathbb {F}}_2^n)^s}-\{0\}$ with small NRT metric, which can be formalized by the notion of designs by Delsarte, as mentioned above. See [13] for analysis of the digital nets via association schemes. Our study is different in that we treat the cases where the $\partial _\rho (Y)/\dim \rho =|I(E_\rho ;Y)|/\dim \rho$ are independent of $\rho \ne 1_G$, while the above studies treat the cases where $|I(E_\rho ;Y)|=0$ holds for some “important” characters $\rho$.