Let φ : ℝ → ℝ be a continuously differentiable function on a finite interval J ⊂ ℝ, and let α = (α1, α2) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α1, α2 is of degree ≤ n and height ≤ Q. Denote by \( {M}_{\varphi}^n\left(Q,\gamma, J\right) \) the set of points α such that |φ(α1) − α2| ≤ c 1 Q −γ. We show that for 0 < γ < 1 and any sufficiently large Q there exist positive values c2 < c3, where ci = ci(n), i = 1, 2, that are independent of Q and such that \( {c}_2\cdot {Q}^{n+1-\upgamma}<\#{M}_{\varphi}^n\left(Q,\upgamma, J\right)<{c}_3\cdot {Q}^{n+1-\upgamma}. \) Bibliography: 17 titles.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 448, 2016, pp. 14–47.
Supported by SFB-701, Bielefeld University (Germany).
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Bernik, V., Gӧtze, F. & Gusakova, A. On the Distribution of Points with Algebraically Conjugate Coordinates in a Neighborhood of Smooth Curves. J Math Sci 224, 176–198 (2017). https://doi.org/10.1007/s10958-017-3404-6
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DOI: https://doi.org/10.1007/s10958-017-3404-6