The sequence of fractional parts of roots

• Autorzy: Kevin O'Bryant
• Języki publikacji: 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.

Kategorie tematyczne

• 11B83: Special sequences and polynomials
• 11J70: Continued fractions and generalizations
• 11J99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 169
• Numer: 4
• Strony: 357-371

Daty

wydano

2015

Twórcy

autor

Kevin O'Bryant

• 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.

Identyfikatory

DOI

10.4064/aa169-4-4