On permutations of Hardy-Littlewood-Pólya sequences

Abstract

Abstract. Let H = (q 1 , . . . , q r ) be a finite set of coprime integers and let n 1 , n 2 , . . . denote the multiplicative semigroup generated by H and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory, and they have remarkable probabilistic and ergodic properties. In particular, the asymptotic properties of the sequence {n k x} are similar to those of independent, identically distributed random variables; here {·} denotes fractional part. In this paper we prove that under mild assumptions on the periodic function f , the sequence f (n k x) obeys the central limit theorem and the law of the iterated logarithm after any permutation of its terms. Note that the permutational invariance of the CLT and LIL generally fails for lacunary sequences f (m k x) even if (m k ) has Hadamard gaps. Our proof depends on recent deep results of Amoroso and Viada on Diophantine equations. We will also show that {n k x} satisfies a strong independence property ("interlaced mixing"), enabling one to determine the precise asymptotic behavior of permuted sums S N (σ) = N k=1 f (n σ(k) x).

On the Uniform Theory of Lacunary Series

“…The purpose of the present chapter is to provide a complete solution of the permutation-invariance of such results. For the proofs we refer to Aistleitner, Berkes and Tichy [2], [3], [5], [6]. Note that for the unpermuted CLT and LIL we need much weaker gap conditions than (1.1).…”

“…Moreover, S (2) N is a nonrandom trigonometric sum, asymptotically independent of S (1) N , whose asymptotic distribution depends sensitively of the gaps ∆ k between the blocks I k and which can be nongaussian. …”

Metric discrepancy results for alternating geometric progressions

“…Before closing introduction, we mention results relating to permutations of sequences. In [17] it was found that the limsups are not invariant under permutations of sequences, and this phenomenon is studied extensively by Aistleitner-Berkes-Tichy [5][6][7][8][9].…”

On the asymptotic behavior of weakly lacunary series

“…Letting · and · p denote the L 2 (0, 1), resp. L p (0, 1) norms, respectively, (17) yields for any positive integer n (r(nx) 2…”

“…Even the number theoretic conditions implying the CLT and LIL under subexponential gap conditions do not help here: the sequence (n k ) k≥1 in Theorem 1 can be chosen so that it satisfies conditions B, C, G in our paper [7] implying very strong independence properties of cos 2πn k x, sin 2πn k x, including the CLT and LIL. In fact, it is very difficult to construct subexponential sequences (n k ) k≥1 satisfying the permutation-invariant CLT and LIL: the only known example (see [2]) is the Hardy-Littlewood-Pólya sequence, i.e. the sequence generated by finitely many primes and arranged in increasing order; the proof uses deep number theoretic tools.…”