Nets in groups, minimum length g -adic representations, and minimal additive complements

📝 Abstract

The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,…} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.

💡 Analysis

The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,…} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.

📄 Content

is an r-net C such that no proper subset of C is an r-net in (X, d). For example, X is a minimal 0-net in (X, d).

Problem 1. In which metric spaces do there exist minimal r-nets for r > 0?

The metric spaces (X, d X ) and (Y, d Y ) are called bi-Lipschitz equivalent if there exists a function f : X → Y such that, for positive constants K 1 and K 2 , we have

for all x, x ′ ∈ X. The metric spaces (X, d X ) and (Y, d Y ) are called quasi-isometric if there exist nets C X in X and C Y in Y that are bi-Lipschitz equivalent. These are fundamental concepts in metric geometry.

Let G be a multiplicative group or semigroup with identity e. For subsets A and B of G, we define the product set AB = {ab : a ∈ A and b ∈ B}. For every nonnegative integer h, we define the product sets A h inductively: A 0 = {e}, A 1 = A, and A h = A h-1 A for h ≥ 2. Thus,

If e ∈ A, then A i-1 ⊆ A i for all i ≥ 1, and

Let A be a set of generators for a group G. Without loss of generality we can assume that A is symmetric, that is, a ∈ A if and only if a -1 ∈ A. We define the word length function ℓ A : G → N 0 as follows: For x ∈ G and x = e, let ℓ A (x) = r if r is the smallest positive integer such that there exist a 1 , a 2 , . . . , a r ∈ A with x = a 1 a 2 • • • a r . Let ℓ A (e) = 0. The integer ℓ A (x) is called the word length of x with respect to A, or, simply, the length of x.

Let A be a symmetric generating set for G. The following properties follow immediately from the definition of the word length function:

(i) ℓ A (x) = 0 if and only if x = e, (ii) ℓ A (x -1 ) = ℓ A (x) for all x ∈ G, (iii) ℓ A (xy) ≤ ℓ A (x) + ℓ A (y) for all x, y ∈ G, (iv) ℓ A (x) = 1 if and only if x ∈ A \ {e}, (v) if x = a 1 • • • a s with a i ∈ A for i = 1, . . . , s, then ℓ A (x) ≤ s, (vi) If A ′ = A ∪ {e}, then ℓ A ′ (x) = ℓ A (x) for all x ∈ G. Lemma 1. Let A be a symmetric generating set for a group G. Suppose that ℓ A (x) = r and that the elements a 1 , a 2 , . . . , a r ∈ A satisfy x = a 1 a 2 • • • a r . For

Proof. By word length properties (iii) and (v) we have

and so

This completes the proof.

Let A be a symmetric generating set for a group G. The length function ℓ A induces a metric d A on G as follows:

The distance between distinct elements of G is always a positive integer, and so the metric space (G, d A ) is 1-separated. Moreover, d A (x, e) = ℓ A (x) for all x ∈ G, and so, for every nonnegative integer h, we have

Thus, the set of all group elements of length h is precisely the sphere with center e and radius h in the metric space (G, d A ).

If r ≥ 0 and h = [r] is the integer part of r, then for every z ∈ G we have

and so the geometry of the group G is determined by closed balls with integer radii. If e ∈ A, then

Theorem 1. Let G be a group and let A be a symmetric generating set for G with e ∈ A. For every nonnegative integer h, the set C is an h-net in the metric space

Thus, C is a net if and only if G = A h C for some nonnegative integer h.

Here are two constructions of nets.

Theorem 2. Let G be a group and let A be a symmetric generating set for G with e ∈ A. For every nonnegative integer h, the set

By the division algorithm, there exist integers q ≥ 0 and r such that

There exist elements a 1 , . . . , a n ∈ A such that

Since this is a shortest representation of x as a product of elements of A, it follows from Lemma 1 that

This completes the proof.

Theorem 3. Let G be a group and let A be a symmetric generating set for G with e ∈ A. Suppose that for every x ∈ G there exists a ∈ A with

For every nonnegative integer h, the set

. By the division algorithm, there exist integers q ≥ 0 and r such that n = r + (2h + 1)q and |r| ≤ h.

If r ≥ 0, then the argument in the proof of Theorem 2 shows that x ∈ A h C. Suppose that r < 0. Then n = (2h + 1)q -|r| and there exist elements a |r|+1 , . . . , a (2h+1)q ∈ A such that

Condition (1) implies that there exist elements a 1 , . . . , a |r| ∈ A such that

This completes the proof.

and Cy is an h-net in G. Thus, the set of h-nets in the metric space (G, d A ) is closed with respect to supersets and right translations. We modify the definitions appropriately when G is an additive abelian group with identity element 0. For subsets A and B of G, we define the sumset

We define 0A = {0}. For every b ∈ G, there is the translation A + b = A + {b}. Let A be a symmetric generating set for G with 0 ∈ A. By Theorem

Let ℓ g and d g denote, respectively, the word length function and the metric induced on Z. Classify the nets in the metric space (Z, d g ). Does this space contain minimal nets? The metrics d 2 and d 3 are particularly interesting.

Fix an integer g ≥ 2, and consider the additive group Z with generating set A g = {0} ∪ {±g i : i = 0, 1, 2, . . .}. We denote by ℓ g (n) the word length of an integer n with respect to A g . A partition of an integer n as a sum of not necessarily distinct elements of A g will be called a g-adic representation of n. In order to understand the me

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