论文信息 - What Is “Probabilistic” Diophantine Approximation?

What Is “Probabilistic” Diophantine Approximation?

We discuss some surprising new developments concerning \(\sqrt{ 2}\), and in general the class of quadratic irrationals. We use \(\sqrt{ 2}\) as the representative for the whole class. These results provide some rigorous evidence for a mysterious general phenomenon that we call the Giant Leap. In a nutshell, it is about the unexpected randomness of explicit sequences (Giant Leap to full-blown randomness). The reader may jump ahead to Sect. 2.5 for a detailed discussion of this issue.

József Beck | J. Beck

[1] William Feller,et al. An Introduction to Probability Theory and Its Applications , 1967 .

[2] J. Beck,et al. Irregularities of distribution , 1987 .

[3] J. Littlewood,et al. Some Problems of Diophantine Approximation: The Lattice-Points of a Right-Angled Triangle , 1922 .

[4] William Feller,et al. An Introduction to Probability Theory and Its Applications , 1951 .

[5] William W. L. Chen. On irregularities of distribution. , 1980 .

[6] G. H. Hardy,et al. Some problems of Diophantine approximation: The lattice-points of a right-angled triangle. (Second memoir.) , 1922 .

[7] Bernard Chazelle,et al. The Discrepancy Method , 1998, ISAAC.

[8] S. Lang,et al. Introduction to Diophantine Approximations , 1995 .

[9] W. Schmidt. IRREGULARITIES OF DISTRIBUTION (Cambridge Tracts in Mathematics 89) , 1988 .

[10] E. Wright,et al. An Introduction to the Theory of Numbers , 1939 .

[11] H. Davenport. Note on irregularities of distribution , 1956 .

[12] A. Ostrowski. Bemerkungen zur Theorie der Diophantischen Approximationen , 1922 .

[13] K. F. Roth. On irregularities of distribution , 1954 .

[14] H. Weyl. Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[15] W. Schmidt. On irregularities of distribution vii , 1972 .

[16] József Beck,et al. Irregularities of distribution: Index of theorems and corollaries , 1987 .

[17] S. Swierczkowski. On successive settings of an arc on the circumference of a circle , 1958 .