### Fraenkel's partition and Brown's decomposition.

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We study the function ${M}_{\theta}\left(n\right)=\lfloor 1/{\theta}^{1/n}\rfloor $, 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)=\lfloor n/log\theta -1/2\rfloor $. 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)=\lfloor n/log\theta -1/2\rfloor $. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...

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 ${G}^{\left(k\right)}={\bigcup}_{i=1}^{\infty}({a}_{2i},{a}_{2i-1}]$ 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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