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## Acta Arithmetica

2015 | 169 | 4 | 357-371
Tytuł artykułu

### The sequence of fractional parts of roots

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EN
Abstrakty
EN
We study the function $M_{θ}(n) = ⌊1/{θ^{1/n}}⌋$, where θ is a positive real number, ⌊·⌋ and {·} are the floor and fractional part functions, respectively. Nathanson proved, among other properties of $M_{θ}$, that if log θ is rational, then for all but finitely many positive integers n, $M_{θ}(n) = ⌊n/log θ - 1/2⌋$. We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy $M_{θ}(n) = ⌊n/log θ - 1/2⌋$. Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued fractions, we produce uncountably many θ that have only finitely many exceptional n, and also give uncountably many explicit θ that have infinitely many exceptional n.
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Tom
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Strony
357-371
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wydano
2015
Twórcy
autor
• Department of Mathematics, College of Staten Island (CUNY), Staten Island, NY 10314, U.S.A.
• CUNY Graduate Center, New York, NY 10016, U.S.A.
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