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Almost-linear segments of graphs of functions. (English. Russian original) Zbl 07179655
Math. Notes 106, No. 5, 720-726 (2019); translation from Mat. Zametki 106, No. 5, 679-686 (2019).
Summary: Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function whose graph \(\{(x, f(x))\}_{x \in \mathbb{R} }\) in \(\mathbb{R}^2\) is a rectifiable curve. It is proved that, for all \(L < \infty\) and \(\varepsilon > 0\), 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 \(( x, f (x))\), \(a \leq x \leq b\), to the segment \(AB\) do not exceed \(\varepsilon|AB|\). 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 \(r < \infty \), there exists a straight line containing at least \(r\) points of this sequence.
MSC:
26 Real functions
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References:
[1] Sagan, H., Space-Filling Curves (1994), New York: Springer-Verlag, New York · Zbl 0806.01019
[2] Schmidt, W., Diophantine Approximation (1980), Heidelberg: Springer-Verlag, Heidelberg · Zbl 0421.10019
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