Zubkov, A. M.; Orlov, O. P. 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 Keywords:rectifiable curve; graph of function; discrete geometry PDF BibTeX XML Cite \textit{A. M. Zubkov} and \textit{O. P. Orlov}, Math. Notes 106, No. 5, 720--726 (2019; Zbl 07179655); translation from Mat. Zametki 106, No. 5, 679--686 (2019) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.