Floor, ceiling and the space between

[1] V. Bergelson, A. Leibman, Distribution of values of bounded generalized polynomials, Acta Math. 198 (2007), no. 2, 155-230. Search in Google Scholar

[2] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Mathematical Physics, no. 35, Cambridge University Press, 1957. Search in Google Scholar

[3] A. Ciubotaru, Problema 27327, Gazeta MatematicȈa, Seria B, no. 1 (2017). Search in Google Scholar

[4] B.Ćurgus, Wolfram Mathematica Explorations for Floor, Ceiling and the Space Between. https://faculty.curgus.wwu.edu/Papers/FloorCeiling.html Search in Google Scholar

[5] A. S. Fraenkel, Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups, Trans. Amer. Math. Soc. 341, no. 2 (1994), 639-664. Search in Google Scholar

[6] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed., Addison-Wesley, 1994. Search in Google Scholar

[7] K. E. Iverson, A Programming Language, John Wiley and Sons, 1962. Search in Google Scholar

[8] D. E. Knuth, The Art of Computer Programming, Addison-Wesley, 1972. Search in Google Scholar

[9] T. Kulhanek, K. McDonough, V. Ponomarenko, Dilated floor functions that commute sometimes, PUMP J. Undergrad. Res. 2 (2019), 107-117. Search in Google Scholar

[10] J. C. Lagarias, T. Murayama, D. H. Richman, Dilated floor functions that commute, Amer. Math. Monthly 123, no. 10 (2016), 1033-1038. Search in Google Scholar

[11] J. C. Lagarias, D. H. Richman, Dilated floor functions having nonnegative commutator I. Positive and mixed sign dilations, Acta Arith. 187, no. 3 (2019), 271-299. Search in Google Scholar

[12] J. C. Lagarias, D. H. Richman, Dilated floor functions having nonnegative commutator II. Negative dilations, Acta Arith. 196, no. 2 (2020), 163-186. Search in Google Scholar

[13] G. Malajovich, E ective versions of Kronecker’s approximation theorem, 2001. https://dma.im.ufrj.br/gregorio/papers/kron.pdf Search in Google Scholar

[14] E. W. Weisstein, Kronecker’s Approximation Theorem, From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/KroneckersApproximationTheorem.html Search in Google Scholar