The metric theory of the pair correlation function of real-valued lacunary sequences

Abstract

Let { a ( x ) } x = 1 ∞ be a positive, real-valued, lacunary sequence. This note shows that the pair correlation function of the fractional parts of the dilations α a ( x ) is Poissonian for Lebesgue almost every α ∈ R . By using harmonic analysis, our result—irrespective of the choice of the real-valued sequence { a ( x ) } x = 1 ∞ —can essentially be reduced to showing that the number of solutions to the Diophantine inequality | n 1 ( a ( x 1 ) − a ( y 1 ) ) − n 2 ( a ( x 2 ) − a ( y 2 ) ) | < 1 in integer six-tuples ( n 1 , n 2 , x 1 , x 2 , y 1 , y 2 ) located in the box [ − N , N ] 6 with the “excluded diagonals”; that is, x 1 ≠ y 1 , x 2 ≠ y 2 , ( n 1 , n 2 ) ≠ ( 0 , 0 ) , is at most N 4 − δ for some fixed δ > 0 , for all sufficiently large N .

Locations

• Illinois Journal of Mathematics - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View