On the law of the iterated logarithm for the discrepancy of Lacunary sequences.
On the law of the iterated logarithm for the discrepancy of Lacunary sequences.
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关于缺位序列差异的迭代对数定律。
DOI：
10.1090/s0002-9947-2012-05740-0
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发表时间：
2010
影响因子：
1.3
通讯作者：
C. Aistleitner
中科院分区：
文献类型：
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作者：
C. Aistleitner
Philipp（1975）的一个经典结果是：对于任意满足Hadamard间隙条件nk +1 /nk ≥ q > 1（k = 1，2，.）的整数序列（nk）k ≥1，序列（nkx）k≥1 mod 1的偏差DN满足重对数律（LIL），即1/4 ≤ limsupNDN（nkx）（NloglogN）-12 ≤ Cqa.e. N→∞ lim sup的值是一个长期存在的公开问题。最近，Fukuyama明确计算了lim sup的值，其中n k = θ k，θ > 1，不一定是整数。我们扩展福山的结果一大类的整数序列（n k）的特点是在一定的一类丢番图方程的解决方案的数量，并表明，该值的lim sup是相同的，在Chung-Smirnov LIL为i.i.d.随机变量
A classical result of Philipp (1975) states that for any sequence (n k ) k ≥1 of integers satisfying the Hadamard gap condition n k+1 /n k ≥ q > 1 (k = 1,2,...), the discrepancy D N of the sequence (n k x)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim sup ND N (n k x)(N log log N) -12 ≤ C q a .e. N→∞ The value of the lim sup is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the lim sup for n k = θ k , θ > 1, not necessarily integer. We extend Fukuyama's result to a large class of integer sequences (n k ) characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the lim sup is the same as in the Chung-Smirnov LIL for i.i.d. random variables.