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Pair Correlations and Random Walks on the Integers

REFERENCES [KN] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences . Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. [NP] NAIR, R.—POLLICOTT, M.: On pair correlations of sequences in higher dimensions , Israel J. Math. 157 (2007), no. 1, 219–238. [Sp] SPITZER, F. L.: Principles of Random Walks . Second ed. Graduate Texts in Mathematics, Vol. 34. Springer-Verlag, New York-Heidelberg, 1976. [St] STONE, C. T.: On local and ratio limit theorems , in: Proc. of the Fifth

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On the Pseudorandomness of the Liouville Function of Polynomials over a Finite Field

REFERENCES [1] BRANDSTÄTTER, N.—WINTERHOF, A.: Linear complexity profile of binary sequences with small correlation measure , Period. Math. Hungar. 52 (2006), no. 2, 1–8. [2] CARLITZ, L.: The arithmetic of polynomials in a Galois field , Amer. J.Math. 54 (1932), no. 1, 39-50. [3] CARMON, D.—RUDNICK, Z.: The autocorrelation of the Möbius function and Chowla’s conjecture for the rational function field , Q. J. Math. 65 (2014), no. 1, 53–61. [4] CASSAIGNE J.—FERENCZI, S.—MAUDUIT, C.—RIVAT, J.—SÁRKÖZY, A.: On finite pseudorandom

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Questions Around the Thue-Morse Sequence

Abstract

We intend to unroll the surprizing properties of the Thue-Morse sequence with a harmonic analysis point of view, and mention in passing some related open questions.

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A Note on the Continued Fraction of Minkowski

Abstract

Denote by Θ12, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

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Discrepancy Results for The Van Der Corput Sequence

.—PILLICHSHAMMER, F.: Sums of distances to the nearest integer and the discrepancy of digital nets , Acta Arith. 106 (2003), no. 4, 379–408. [16] LEHMER, D. H.: On Stern’s Diatomic Series , Amer. Math. Monthly 36 (1929), no. 2, 59–67. [17] LIND, D. A.: An extension of Stern’s diatomic series , Duke Math. J. 36 (1969), 55–60. [18] MORGENBESSER, J. F.—SPIEGELHOFER, L.: A reverse order property of correlation measures of the sum-of-digits function , Integers, 12 (2012), Paper No. A47. [19] F. PILLICHSHAMMER, F.: On the discrepancy of (0, 1

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Pierre Liardet (1943–2014)

for binary strings with low aperiodic auto-correlations , Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), Vol. 2936, Springer-Berlin, 2004, pp. 39–50. [L41] B arat , G.—L iardet , P.: Dynamical systems originated in the Ostrowski alpha-expansion , Ann. Univ. Sci. Budapest. Sect. Comput. 24 (2004), 133–184. [L42] G rabner , P. J.—L iardet , P.—T ichy , R. F.: Spectral disjointness of dynamical systems related to some arithmetic functions , Publ. Math. Debrecen 66

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