PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 3, March 2012, Pages 749–754 S 0002-9939(2011)10940-7 Article electronically published on June 23, 2011 A METRIC DISCREPANCY RESULT FOR LACUNARY SEQUENCES KATUSI FUKUYAMA AND TETSUJIN WATADA (Communicated by Richard C. Bradley) Abstract. We prove that every value greater than or equal to 1/2 can be a constant appearing in the law of the iterated logarithm for discrepancies of a lacunary sequence satisfying the Hadamard gap condition. 1. Introduction In the theory of the uniform distribution, we use the following discrepancies of a sequence {a k }: D N {a k } = sup 0≤a ′ <a<1     1 N # {k ≤ N |〈a k 〉∈ [ a ′ ,a)}− (a − a ′ )     , D ∗ N {a k } = sup 0≤a<1     1 N # {k ≤ N |〈a k 〉∈ [0,a)}− a     , where 〈x〉 denotes the fractional part x − [ x ] of x. One of the most typical results on the asymptotic behavior of discrepancies is celebrated in the Chung-Smirnov theorem, which asserts the law of the iterated logarithm for discrepancies of uniformly distributed i.i.d. {U k }: lim N→∞ ND ∗ N {U k } √ 2N log log N = lim N→∞ ND N {U k } √ 2N log log N = 1 2 , a.s. We have similar phenomena without assuming the independence of the sequence of random variables. For a sequence {n k } satisfying the Hadamard gap condition, (1.1) inf k∈N n k+1 /n k > 1, Philipp [9, 10] proved the bounded law of the iterated logarithm 1 4 √ 2 ≤ Σ ∗ {n k x} := lim N→∞ ND ∗ N {n k x} √ 2N log log N ≤ Σ{n k x} := lim N→∞ ND N {n k x} √ 2N log log N ≤ C for almost every x, where C is a constant depending only on the infimum in (1.1). Recently it became possible to calculate concrete values of Σ{n k x} and Σ ∗ {n k x}. Received by the editors November 26, 2010 and, in revised form, December 8, 2010. 2010 Mathematics Subject Classification. Primary 11K38; Secondary 60F15. Key words and phrases. Discrepancy, lacunary sequence, law of the iterated logarithm. The first author was supported in part by KAKENHI 19204008. c 2011 American Mathematical Society Reverts to public domain 28 years from publication 749 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use