Difference bases in dihedral groups

TL;DR

This paper investigates the minimal size of difference bases in dihedral groups, establishing bounds on their difference characteristic and computing exact values for small groups, advancing understanding of their algebraic structure.

Contribution

It provides new bounds on the difference characteristic of dihedral groups and computes exact difference sizes for groups up to order 80.

Findings

For all n, the difference characteristic of D_{2n} is between √2 and approximately 1.983.

For large n (≥ 2×10^{15}), the difference characteristic is less than approximately 1.633.

Exact difference sizes and characteristics are computed for all dihedral groups of order ≤ 80.

Abstract

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is called the difference characteristic of $G$. We prove that for every $n\in\mathbb N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\sqrt{2}\le\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$. Moreover, if $n\ge 2\cdot 10^{15}$, then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$.

Tables

Table 1. Table 1. Difference sizes and characteristics of cyclic groups C n subscript 𝐶 𝑛 C_{n} for n ≤ 100 𝑛 100 n\leq 100

$n$	​$\Delta \left[C_n\right]$​	$ð\left[C_n\right]$	$n$	​$\Delta \left[C_n\right]$​	$ð\left[C_n\right]$	$n$	​$\Delta \left[C_n\right]$​	$ð\left[C_n\right]$	$n$	​$\Delta \left[C_n\right]$​	$ð\left[C_n\right]$
1	1	1	26	6	1.1766…​​	51	8	1.1202…​​	76	10	1.1470…
2	2	1.4142…	27	6	1.1547…​​	52	9	1.2480…​​	77	10	1.1396…​​
3	2	1.1547…	28	6	1.1338…​​	53	9	1.2362…​​	78	10	1.1322…​​
4	3	1.5	29	7	1.2998…​​	54	9	1.2247…​​	79	10	1.1250…​​
5	3	1.3416…	30	7	1.2780…​​	55	9	1.2135…​​	80	11	1.2298…​​
6	3	1.2247…	31	6	1.0776…​​	56	9	1.2026…​​	81	11	1.2222…​​
7	3	1.1338…	32	7	1.2374…​​	57	8	1.0596…​​	82	11	1.2147…​​
8	4	1.4142…	33	7	1.2185…​​	58	9	1.1817…​​	83	11	1.2074…​​
9	4	1.3333…	34	7	1.2004…​​	59	9	1.1717…​​	84	11	1.2001…​​
10	4	1.2649…	35	7	1.1832…​​	60	9	1.1618…​​	85	11	1.1931…​​
11	4	1.2060…	36	7	1.1666…​​	61	9	1.1523…​​	86	11	1.1861…​​
12	4	1.1547…	37	7	1.1507…​​	62	9	1.1430…​​	87	11	1.1793…​​
13	4	1.1094…	38	8	1.2977…​​	63	9	1.1338…​​	88	11	1.1726…​​
14	5	1.3363…	39	7	1.1208…​​	64	9	1.125​​	89	11	1.1659…​​
15	5	1.2909…	40	8	1.2649…​​	65	9	1.1163…​​	90	11	1.1595…​​
16	5	1.25	41	8	1.2493…​​	66	10	1.2309…​​	91	10	1.0482…​​
17	5	1.2126…	42	8	1.2344…​​	67	10	1.2216…​​	92	11	1.1468…​​
18	5	1.1785…	43	8	1.2199…​​	68	10	1.2126…​​	93	12	1.2443…​​
19	5	1.1470…	44	8	1.2060…​​	69	10	1.2038…​​	94	12	1.2377…​​
20	6	1.3416…	45	8	1.1925…​​	70	10	1.1952…​​	95	12	1.2311…​​
21	5	1.0910…	46	8	1.1795…​​	71	10	1.1867…​​	96	12	1.2247…​​
22	6	1.2792…	47	8	1.1669…​​	72	10	1.1785…​​	97	12	1.2184…​​
23	6	1.2510…	48	8	1.1547…​​	73	9	1.0533…​​	98	12	1.2121…​​
24	6	1.2247…	49	8	1.1428…​​	74	10	1.1624…​​	99	12	1.2060…​​
25	6	1.2	50	8	1.1313…​​	75	10	1.1547…​​	100	12	1.2

Table 2. Table 2. Difference sizes and characteristics of dihedral groups D 2 ​ n subscript 𝐷 2 𝑛 D_{2n} for 2 ​ n ≤ 80 2 𝑛 80 2n\leq 80 .

$2n$	​$lb\left[D_{2n}\right]$​	$\Delta \left[D_{2n}\right]$	$2\Delta \left[C_n\right]$	​$ð\left[D_{2n}\right]$	$2n$	​$lb\left[D_{2n}\right]$​	$\Delta \left[D_{2n}\right]$	$2\Delta \left[C_n\right]$	​$ð\left[D_{2n}\right]$
2	2	2	2	1.4142…​​	42	10	10	10	1.5430…​​
4	3	3	4	1.5​​	44	10	10	12	1.5075…​​
6	4	4	4	1.6329…​​	46	10	11	12	1.6218…​​
8	4	4	6	1.4142…​​	48	10	10	12	1.4433…​​
10	5	5	6	1.5811…​​	50	10	11	12	1.5556…​​
12	5	5	6	1.4433…​​	52	11	11	12	1.5254…​​
14	6	6	6	1.6035…​​	54	11	12	12	1.6329…​​
16	6	6	8	1.5​​	56	11	11	12	1.4699…​​
18	6	7	8	1.6499…​​	58	11	12	14	1.5756…​​
20	7	7	8	1.5652…​​	60	11	12	14	1.5491…​​
22	7	8	8	1.7056…​​	62	12	12	12	1.5240…​​
24	7	7	8	1.4288…​​	64	12	12	14	1.5​​
26	8	8	8	1.5689…​​	66	12	13	14	1.6001…​​
28	8	8	10	1.5118…​​	68	12	13	14	1.5764…​​
30	8	8	10	1.4605…​​	70	12	12	14	1.4342…​​
32	8	9	10	1.5909…​​	72	12	13	14	1.5320…​​
34	9	9	10	1.5434…​​	74	13	14	14	1.6274…​​
36	9	9	10	1.5​​	76	13	14	16	1.6059…​​
38	9	10	10	1.6222…​​	78	13	14	14	1.5851…​​
40	9	9	12	1.4230…​​	80	13	14	16	1.5652…​​

Equations

$$ð[A]:=\frac{\Delta [A]}{\sqrt{|A|}}\ge 1$$

$$\lceil x\rceil =\min \{n\in \mathbb{Z}:n\ge x\}\text{mbox}and\lfloor x\rfloor =\max \{n\in \mathbb{Z}:n\le x\}.$$

$$C_n=\{z\in \mathbb{C}:z^n=1\}$$

$$2\sqrt{nm}\le \Delta [D_{2nm}]\le \Delta [D_{2n}]\cdot \Delta [C_m]$$

$$x\in A{B}^{-1}{s}^{-1}{s}^{-1}\cup sB{A}^{-1}{s}^{-1}=A{B}^{-1}\cup B^{-1}A=A{B}^{-1}.$$

$$2\sqrt{n}\le \Delta [D_{2n}]\le 2\cdot \Delta [C_n]$$

$$\sqrt{2+\frac{4}{3\pi }}<\sqrt{2+\underset{0<\phi <2\pi }{\max }\frac{2\sin (\phi )}{\phi +\pi }}\le \underset{n\to \infty }{\lim }ð[n]=\underset{n\in \mathbb{N}}{\inf }ð[n]\le ð[6166]=\frac{128}{\sqrt{6166}}<ð[6]=\sqrt{\frac{8}{3}}.$$

$$\underset{n\to \infty }{\mathrm{limsup}}ð[C_n]\le \frac{1}{\sqrt{2}}\underset{n\in \mathbb{N}}{\inf }ð[n]\le \frac{64}{\sqrt{3083}}<\frac{2}{\sqrt{3}}.$$

$$\Delta[D_{2n}]=2\cdot\Delta[C_{n}]=\left\lceil 2\sqrt{n}\,\right\rceil=\Big{\lceil}\sqrt{2|D_{2n}|}\,\Big{\rceil}=2+2q.$$

$$2\sqrt{q^2+q+1}=2\sqrt{n}\le \Delta [D_{2n}]\le \Delta [D_2]\cdot \Delta [C_n]=2\cdot \Delta [C_n]=2+2q,$$

$$4q+3\le \Delta [D_{8n}]\le 4q+4.$$

$$4\sqrt{q^2+q+1}=2\sqrt{4n}\le \Delta [D_{8n}]\le \Delta [D_8]\cdot \Delta [C_n]=4(1+q).$$

$$\sqrt{2}\le ð[D_{2n}]\le \frac{48}{\sqrt{586}}\approx \mathrm{1.983.}$$

$$ð[D_{2\cdot 292}]=ð[D_{8\cdot 73}]\le ð[D_8]\cdot ð[C_{73}]=\sqrt{2}\cdot \frac{9}{\sqrt{73}}<\frac{48}{\sqrt{586}}.$$

$$ð[D_{2n}]\le \sqrt{2}\cdot ð[C_n]<\frac{4}{\sqrt{6}}.$$

$$4(q_{k+1}+1)\ge \Delta [C_n]>\frac{4}{\sqrt{11}}\sqrt{n}\ge \frac{4}{\sqrt{11}}\sqrt{12q_k^2+14q_k+16}.$$

Full text

A subset $B$ of a group $G$ is called a difference basis for a subset $A\subset G$ if each element $a\in A$ can be written as $a=xy^{-1}$ for some $x,y\in B$. The smallest cardinality of a difference basis for $A$ is called the difference size of $A$ and is denoted by $\Delta[A]$. For example, the set $\{0,1,4,6\}$ is a difference basis for the interval $A=[-6,6]\cap\mathbb{Z}$ witnessing that $\Delta[A]\leq 4$.

The definition of a difference basis $B$ for a set $A$ in a group $G$ implies that $|A|\leq|B|^{2}$ and gives a lower bound $\sqrt{|A|}\leq\Delta[A]$. The fraction

[TABLE]

is called the difference characteristic of $A$.

For a real number $x$ we put

[TABLE]

The following proposition is proved in [1, 1.1].

1. (1)

\frac{1+\sqrt{4|G|-3}}{2}\leq\Delta[G]\leq\big{\lceil}\frac{|G|+1}{2}\big{\rceil}, 2. (2)

$\Delta[G]\leq\Delta[H]\cdot\Delta[G/H]$* and $\eth[G]\leq\eth[H]\cdot\eth[G/H]$ for any normal subgroup $H\subset G$;* 3. (3)

$\Delta[G]\leq|H|+|G/H|-1$* for any subgroup $H\subset G$.*

In [8] Kozma and Lev proved (using the classification of finite simple groups) that each finite group $G$ has difference characteristic $\eth[G]\leq\frac{4}{\sqrt{3}}\approx 2.3094$.

In this paper we shall evaluate the difference characteristics of dihedral groups and prove that each diherdal group $D_{2n}$ has $\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx 1.983$. Moreover, if $n\geq 2\cdot 10^{15}$, then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx 1.633$. We recall that the dihedral group $D_{2n}$ is the isometry group of a regular $n$-gon. The dihedral group $D_{2n}$ contains a normal cyclic subgroup of index 2. A standard model of a cyclic group of order $n$ is the multiplicative group

[TABLE]

of $n$-th roots of $1$. The group $C_{n}$ is isomorphic to the additive group of the ring $\mathbb{Z}_{n}=\mathbb{Z}/n\mathbb{Z}$.

[TABLE]

and the difference characteristic $\sqrt{2}\leq\eth[D_{2nm}]\leq\eth[D_{2n}]\cdot\eth[C_{m}]$.

Next, we prove the lower bound $2\sqrt{nm}\leq\Delta[D_{2nm}]$. Fix any element $s\in D_{2nm}\setminus C_{nm}$ and observe that $s=s^{-1}$ and $sxs^{-1}=x^{-1}$ for all $x\in C_{nm}$. Fix a difference basis $D\subset D_{2nm}$ of cardinality $|D|=\Delta[D_{2nm}]$ and write $D$ as the union $D=A\cup sB$ for some sets $A,B\subset C_{nm}\subset D_{2nm}$. We claim that $AB^{-1}=C_{nm}$. Indeed, for any $x\in C_{nm}$ we get $xs\in sC_{nm}\cap(A\cup sB)(A\cup sB)^{-1}=AB^{-1}s^{-1}\cup sBA^{-1}$ and hence

[TABLE]

So, $C_{nm}=AB^{-1}$ and hence $nm\leq|A|\cdot|B|$. Then $\Delta[D_{2nm}]=|A|+|B|\geq\min\{l+k:l,k\in\mathbb{N},\;lk\geq nm\}\geq 2\sqrt{nm}$ and $\eth[D_{2nm}]=\frac{\Delta[D_{2nm}]}{\sqrt{2nm}}\geq\frac{2\sqrt{nm}}{\sqrt{2nm}}=\sqrt{2}$. ∎

[TABLE]

and the difference characteristic $\sqrt{2}\leq\eth[D_{2n}]\leq\sqrt{2}\cdot\eth[C_{n}]$.

The difference sizes of finite cyclic groups were evaluated in [2] with the help of the difference sizes of the order-intervals $[1,n]\cap\mathbb{Z}$ in the additive group $\mathbb{Z}$ of integer numbers. For a natural number $n\in\mathbb{N}$ by $\Delta[n]$ we shall denote the difference size of the order-interval $[1,n]\cap\mathbb{Z}$ and by $\eth[n]:=\frac{\Delta[n]}{\sqrt{n}}$ its difference characteristic. The asymptotics of the sequence $(\eth[n])_{n=1}^{\infty}$ was studied by Rédei and Rényi [9], Leech [7] and Golay [6] who eventually proved that

[TABLE]

In [2] the difference sizes of the order-intervals $[1,n]\cap\mathbb{Z}$ were applied to give upper bounds for the difference sizes of finite cyclic groups.

[TABLE]

The following upper bound for the difference sizes of cyclic groups were proved in [2].

1. (1)

$\eth[C_{n}]\leq\eth[C_{4}]=\frac{3}{2}$; 2. (2)

$\eth[C_{n}]\leq\eth[C_{2}]=\eth[C_{8}]=\sqrt{2}$* if $n\neq 4$;* 3. (3)

$\eth[C_{n}]\leq\frac{12}{\sqrt{73}}<\sqrt{2}$* if $n\geq 9$;* 4. (4)

$\eth[C_{n}]\leq\frac{24}{\sqrt{293}}<\frac{12}{\sqrt{73}}$* if $n\geq 9$ and $n\neq 292$;* 5. (5)

$\eth[C_{n}]<\frac{2}{\sqrt{3}}$* if $n\geq 2\cdot 10^{15}$.*

For some special numbers $n$ we have more precise upper bounds for $\Delta[C_{n}]$. A number $q$ is called a prime power if $q=p^{k}$ for some prime number $p$ and some $k\in\mathbb{N}$.

The following theorem was derived in [2] from the classical results of Singer [11], Bose, Chowla [3], [4] and Rusza [10].

1. (1)

$\Delta[C_{q^{2}+q+1}]=q+1$; 2. (2)

$\Delta[C_{q^{2}-1}]\leq q-1+\Delta[C_{q-1}]\leq q-1+\frac{3}{2}\sqrt{q-1}$; 3. (3)

$\Delta[C_{p^{2}-p}]\leq p-3+\Delta[C_{p}]+\Delta[C_{p-1}]\leq p-3+\frac{3}{2}(\sqrt{p}+\sqrt{p-1})$.

The following Table 1 of difference sizes and characteristics of cyclic groups $C_{n}$ for $\leq 100$ is taken from [2].

Using Theorem 6(1), we shall prove that for infinitely many numbers $n$ the lower and upper bounds given in Theorem 2 uniquely determine the difference size $\Delta[D_{2n}]$ of $D_{2n}$.

[TABLE]

[TABLE]

it suffices to check that $(2+2q)-2\sqrt{q^{2}+q+1}<1$, which is equivalent to $\sqrt{q^{2}+q+1}>q+\frac{1}{2}$ and to $q^{2}+q+1>q^{2}+q+\frac{1}{4}$. ∎

A bit weaker result holds also for the dihedral groups $D_{8(q^{2}+q+1)}$.

[TABLE]

[TABLE]

To see that $4q+3\leq\Delta[D_{8n}]\leq 4q+4$, it suffices to check that $(4+4q)-4\sqrt{q^{2}+q+1}<2$, which is equivalent to $\sqrt{q^{2}+q+1}>q+\frac{1}{2}$ and to $q^{2}+q+1>q^{2}+q+\frac{1}{4}$. ∎

In Table 2 we present the results of computer calculation of the difference sizes and characteristics of dihedral groups of order $\leq 80$. In this table $lb[D_{2n}]:=\lceil\sqrt{4n}\,\rceil$ is the lower bound given in Theorem 2. With the boldface font we denote the numbers $2n\in\{14,26,42,62\}$, equal to $2(q^{2}+q+1)$ for a prime power $q$. For these numbers we know that $\Delta[D_{2n}]=lb[D_{2n}]=2q+2$. For $q=2$ and $n=q^{2}+q+1=7$ the table shows that $\Delta[D_{56}]=\Delta[D_{8n}]=11=4q+3$, which means that the lower bound $4q+3$ in Proposition 8 is attained.

[TABLE]

Moreover, if $n\geq 2\cdot 10^{15}$, then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx 1.633$.

[TABLE]

Analyzing the data from Table 2, one can check that $\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx 1.983$ for all $n\leq 8$.

If $n\geq 2\cdot 10^{15}$, then $\eth[C_{n}]<\frac{2}{\sqrt{3}}$ by Theorem 5(5), and hence

[TABLE]

∎

To answer Question 10 affirmatively, it suffices to check that $\eth[D_{2n}]\leq\frac{8}{\sqrt{22}}$ for all $n<1\,212\,464$.

[TABLE]

By Theorem 1.9 of [5], if $q_{k}\geq 3275$, then $q_{k+1}\leq q_{k}+\frac{q_{k}}{2\ln^{2}(q_{k})}$. On the other hand, using WolframAlpha computational knowledge engine it can be shown that the inequality $1+x+\frac{x}{2\ln^{2}(x)}\leq\frac{1}{\sqrt{11}}\sqrt{12x^{2}+14x+16}$ holds for all $x\geq 43$. This implies that $q_{k}<3275$.

Analysing the table111See https://primes.utm.edu/notes/GapsTable.html and https://primes.utm.edu/lists/small/1000.txt of (maximal gaps between) primes, it can be shows that $11(q_{k+1}+1)^{2}\leq{12q_{k}^{2}+14q_{k}+16}$ if $q_{k}\geq 331$. So, $q_{k}\leq 317$, $q_{k+1}\leq 331$ and $11\cdot(q_{k+1}+1)^{2}=11\cdot 332^{2}=1\,212\,464\leq n$, which contradicts $4(q_{k+1}+1)>\frac{4}{\sqrt{11}}\sqrt{n}$. ∎