Lohöfer, Georg; Mayer, Dieter On a theorem by Florek and Slater on recurrence properties of circle maps. (English) Zbl 0713.11044 Publ. Res. Inst. Math. Sci. 26, No. 2, 335-357 (1990). Let \(\beta\) be real and \(0<a\leq 1\). Then the theorem by Florek and Slater says that the gaps between the successive values of n for which \(n\beta\) mod 1\(<a\) can have at most three lengths [cf. N. B. Slater, Proc. Camb. Philos. Soc. 63, 1115-1123 (1967; Zbl 0178.047)]. The authors show that this result holds also for the gaps of \(-a<n\beta\) mod 1\(<a\). An application is given to the recurrence behaviour of the unit circle under rotations by the angle \(\beta\). Reviewer’s remark: Using only one sheet it can easily be shown that the theorem by Florek and Slater holds for the gaps of \(a\leq n\beta mod 1<b\) for any \(a<b\). Reviewer: P.Schatte Cited in 1 Document MSC: 11K06 General theory of distribution modulo \(1\) 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics Keywords:recurrence times mod 1; formula of Florek and Slater; integrable Hamiltonian systems; ergodic properties of certain chaotic semiflows with strange attractors Citations:Zbl 0178.047 PDFBibTeX XMLCite \textit{G. Lohöfer} and \textit{D. Mayer}, Publ. Res. Inst. Math. Sci. 26, No. 2, 335--357 (1990; Zbl 0713.11044) Full Text: DOI References: [1] Lohofer, G., and Mayer, D., Correlation functions of a time continuous dissipative system with a strange attractor, Phys. Lett., 113A (1985), 105-110. [2] Florek, K., Une remarque sur la repartition des nombres n£mod 1, Coll. Math. Wroclaw, 1 (1951), 323-324. [3] Slater, N. B., Gaps and steps for the sequence nO mod 1, Proc. Comb. Phil. Soc., 63 (1967), 1115-1123. · Zbl 0178.04703 [4] Birkhoff, G., Proof of a recurrence theorem for strongly transitive systems, Proc. Nat. Acad. Sci. USA, 17 (1931), 650-655. · Zbl 0003.25601 · doi:10.1073/pnas.17.12.650 [5] Kac. M., Probability and Related Topics in Physical Sciences, Interscience Publishers LTD, London, 1959. Smoluchowski, R., Drei Vortrage iiber Diffusion, Brown-sche Molekularbewegung und Koagulation von Kolloidteilchen, Phys. Z, 17 (1916), 557-571, 587-599. [6] Slater, N. B., The distribution of the integers N for which {ON} <= 0, Proc. Camb. Phil. Soc., 46 (1950), 525-534. · Zbl 0038.02802 [7] Lohofer, G., Korrelationsfunktionen eines zeitlich kontinuierlichen, chaotischen Systems mit seltsamem Attraktor, Ph.-D Thesis RWTH Aachen (1986), unpublished. · Zbl 0616.58026 [8] Coquet, J., Rhin, G., and Toffin, Ph., Representation des entiers naturels et independence statistique, Ann. Inst. Fourier, Grenoble, 31 (1981), 1-15. · Zbl 0437.10026 · doi:10.5802/aif.814 [9] Lang, S., Introduction to Diophantine approximations, Addison Wesley, London 1966. · Zbl 0144.04005 [10] Khinchin, A. Ya., Continued fractions, Univ. of Chicago Press, Chicago, 1964. · Zbl 0117.28601 [11] Dupain, Y., and Sos, V., On the one-sided boundedness of discrepency function of the sequence {no}, Acta Arithmetica, XXXVII (1980), 363-374. · Zbl 0445.10041 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.