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## Colloquium Mathematicum

2000 | 84/85 | 1 | 125-145
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### Symmetric cocycles and classical exponential sums

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This paper considers certain classical exponential sums as examples of cocycles with additional symmetries. Thus we simplify the proof of a result of Anderson and Pitt concerning the density of lacunary exponential partial sums $\sum_{k=0}^n exp(2πim^{k}x)$, n=1,2,..., for fixed integer m ≥ 2. Also, with the help of Hardy and Littlewood's approximate functional equation, but otherwise by elementary considerations, we improve a previous result of the author for certain examples of Weyl sum: if θ ∈ [0,1] \ ℚ has continued fraction representation $[a_{1},a_{2},... ]$ such that $\sum_{n} 1/a_{n} < ∞$, and $|θ - p/q| < 1/q^{4+ε}$ infinitely often for some ε $#62; 0, then, for Lebesgue almost all x ∈ [0,1], the partial sums$\sum_{k=0}^n exp(2πi(k^{2}θ + 2kx))$, n=1,2,..., are dense in ℂ. Słowa kluczowe Czasopismo Rocznik Tom Numer Strony 125-145 Opis fizyczny Daty wydano 2000 otrzymano 1999-05-25 poprawiono 2000-01-05 Twórcy autor • Department of Mathematics, National University of Ireland, Cork, Republic of Ireland Bibliografia • [AP1] J. M. Anderson and L. D. Pitt, On recurrence properties of certain lacunary series I, J. Reine Angew. Math. 377 (1987), 65-82. • [AP2] J. M. Anderson and L. D. Pitt, On recurrence properties of certain lacunary series II, ibid., 83-96. • [At1] G. Atkinson, Non-compact extensions of transformations, Ph.D. Thesis, Univ. of Warwick, 1976. • [At2] G. Atkinson, Recurrence of cocycles and random walks, J. London Math. Soc. (2) 13 (1976), 486-488. • [At3] G. Atkinson, A class of transitive cylinder transformations, ibid. 17 (1978), 263-270. • [Bi] P. Billingsley, Ergodic Theory and Information, Wiley Ser. Probab. Math. Statist., Wiley, New York, 1965. • [Co] Z. Coelho, On the asymptotic range of cocycles for shifts of finite type, Ergodic Theory Dynam. Systems 13 (1993), 249-262. • [D] F. M. Dekking, On transience and recurrence of generalised random walks, Z. Wahrsch. Verw. Gebiete 61 (1982), 459-465. • [GM-F] F. M. Dekking and M. Mendès-France, Uniform distribution modulo one: a geometrical viewpoint, J. Reine Angew. Math. 329 (1981), 143-153. • [FM] J. Feldman and C. Moore, Ergodic equivalence relations, cohomology and von Neumann algebras I, Trans. Amer. Math. Soc. 234 (1977), 289-324. • [Fo] A. H. Forrest, The limit points of Weyl sums and other continuous cocycles, J. London Math. Soc. (2) 54 (1996), 440-452. • [F] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981. • [G] Y. Guivarc'h, Propriétés ergodiques, en mesure infini, de certains systèmes dynamiques fibrés, Ergodic Theory Dynam. Systems 9 (1989), 433-453. • [HL] G. H. Hardy and J. E. Littlewood, The trigonometric series associated with the elliptic$θ\$-functions, Acta Math. 37 (1914), 193-239.
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