Abstract
Let f: ℝ → ℝ be a function whose graph {(x, f(x))}x∈ℝ in ℝ2 is a rectifiable curve. It is proved that, for all L < ∞ and ɛ > 0, there exist points A = (a, f(a)) and B = (b, f(b)) such that the distance between A and B is greater than L and the distances from all points (x, f(x)), a ≤ x ≤ b, to the segment AB do not exceed ε|AB|. An example of a plane rectifiable curve for which this statement is false is given. It is shown that, given a coordinate-wise nondecreasing sequence of integer points of the plane with bounded distances between adjacent points, for any r < ∞, there exists a straight line containing at least r points of this sequence.
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References
H. Sagan, Space-Filling Curves (Springer-Verlag, New York, 1994).
W. Schmidt, Diophantine Approximation (Springer-Verlag, Heidelberg, 1980; Mir, Moscow, 1983).
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 5, pp. 679–686.
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Zubkov, A.M., Orlov, O.P. Almost-Linear Segments of Graphs of Functions. Math Notes 106, 720–726 (2019). https://doi.org/10.1134/S0001434619110063
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DOI: https://doi.org/10.1134/S0001434619110063