Distribution of real algebraic numbers of arbitrary degree in short intervals

Bernik VI, Götze F (2015)
Izvestiya Mathematics 79(1): 18-39.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
We consider real algebraic numbers alpha of degree deg alpha = n and height H = H(alpha). There are intervals I subset of R of length vertical bar I vertical bar whose interiors contain no real algebraic numbers alpha of any degree with H(alpha) < 1/2 vertical bar I vertical bar(-1). We prove that one can always find a constant c(1) = c(1)(n) such that if Q is a positive integer and Q > c(1)vertical bar I vertical bar(-1), then the interior of I contains at least c(2)(n)Q(n+1) vertical bar I vertical bar real algebraic numbers alpha with deg alpha = n and H(alpha) <= Q. We use this result to solve a problem of Bugeaud on the regularity of the set of real algebraic numbers in short intervals.
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Izvestiya Mathematics
Band
79
Ausgabe
1
Seite(n)
18-39
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Bernik VI, Götze F. Distribution of real algebraic numbers of arbitrary degree in short intervals. Izvestiya Mathematics. 2015;79(1):18-39.
Bernik, V. I., & Götze, F. (2015). Distribution of real algebraic numbers of arbitrary degree in short intervals. Izvestiya Mathematics, 79(1), 18-39. doi:10.1070/IM2015v079n01ABEH002732
Bernik, V. I., and Götze, F. (2015). Distribution of real algebraic numbers of arbitrary degree in short intervals. Izvestiya Mathematics 79, 18-39.
Bernik, V.I., & Götze, F., 2015. Distribution of real algebraic numbers of arbitrary degree in short intervals. Izvestiya Mathematics, 79(1), p 18-39.
V.I. Bernik and F. Götze, “Distribution of real algebraic numbers of arbitrary degree in short intervals”, Izvestiya Mathematics, vol. 79, 2015, pp. 18-39.
Bernik, V.I., Götze, F.: Distribution of real algebraic numbers of arbitrary degree in short intervals. Izvestiya Mathematics. 79, 18-39 (2015).
Bernik, V. I., and Götze, Friedrich. “Distribution of real algebraic numbers of arbitrary degree in short intervals”. Izvestiya Mathematics 79.1 (2015): 18-39.