# Geometric group theory and arithmetic diameter

@article{Nathanson2011GeometricGT, title={Geometric group theory and arithmetic diameter}, author={Melvyn B. Nathanson}, journal={arXiv: Number Theory}, year={2011} }

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let… Expand

#### 12 Citations

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Let P be a finite set of at least two prime numbers and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be… Expand

2 1 D ec 2 01 8 RIGIDITY SEQUENCES , KAZHDAN SETS AND GROUP TOPOLOGIES ON THE INTEGERS by Catalin Badea

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— We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that… Expand

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We use Nathanson's $g$-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets $S$ to problems in additive number… Expand

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We compute metric properties of Cayley graphs of the integers with respect to various infinite generating sets. When the generating set $S$ is the set of all powers of a prime, we find explicit… Expand

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We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity… Expand

Representing integers as sums or differences of general power products

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We extend a result of Hajdu and Tijdeman concerning the smallest number which cannot be obtained as a sum of less than k power products of fixed primes. For this, we also generalize a classical… Expand

Representing integers as linear combinations of powers

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At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by… Expand

Limit Points of Nathanson’s Lambda Sequences

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We consider the set $A_{n}=\displaystyle\cup_{j=0}^{\infty}\{\varepsilon_{j}(n)\cdot n^j\colon\varepsilon_{j}(n)\in\{0,\pm1,\pm2,...,\pm\lfloor{{n}/{2}}\rfloor\}\} $. Let $\mathcal{S}_{\mathcal{A}}=… Expand

Special Representations, Nathanson's Lambda Sequences and Explicit Bounds

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Special Representations, Nathanson’s Lambda Sequences and Explicit Bounds.

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At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by… Expand

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