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Integers

### A complete annotated bibliography of work related to Sidon sequences.

The Electronic Journal of Combinatorics [electronic only]

Integers

### Sets of integers that do not contain long arithmetic progressions.

The Electronic Journal of Combinatorics [electronic only]

### The sequence of fractional parts of roots

Acta Arithmetica

We study the function ${M}_{\theta }\left(n\right)=⌊1/{\theta }^{1/n}⌋$, where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of ${M}_{\theta }$, that if log θ is rational, then for all but finitely many positive integers n, ${M}_{\theta }\left(n\right)=⌊n/log\theta -1/2⌋$. We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy ${M}_{\theta }\left(n\right)=⌊n/log\theta -1/2⌋$. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...

Acta Arithmetica

### A problem of Rankin on sets without geometric progressions

Acta Arithmetica

A geometric progression of length k and integer ratio is a set of numbers of the form $a,ar,...,a{r}^{k-1}$ for some positive real number a and integer r ≥ 2. For each integer k ≥ 3, a greedy algorithm is used to construct a strictly decreasing sequence ${\left({a}_{i}\right)}_{i=1}^{\infty }$ of positive real numbers with a₁ = 1 such that the set ${G}^{\left(k\right)}={\bigcup }_{i=1}^{\infty }\left({a}_{2i},{a}_{2i-1}\right]$ contains no geometric progression of length k and integer ratio. Moreover, ${G}^{\left(k\right)}$ is a maximal subset of (0,1] that contains no geometric progression of length k and integer ratio. It is also proved that there is...

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