On a problem of Bourgain concerning the L¹-norm of exponential sums

2012/11/20 by Christoph Aistleitner, Aistleitner, Christoph
Mathematics · #33B10 #42A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33B10 #msc:42A05

paper · pdf · doi:10.48550/arxiv.1211.4640

arxiv created 2012/11/20 · arxiv updated 2012/11/21

Abstract

Bourgain posed the problem of calculating Σ= sup_{n ≥ 1} ~sup_{k1 <... < kn} (1)/(√(n))‖ ∑_j=1ⁿ e^{2 πi k_j θ}‖_L¹([0,1]). It is clear that Σ≤ 1; beyond that, determining whether Σ< 1 or Σ=1 would have some interesting implications, for example concerning the problem whether all rank one transformations have singular maximal spectral type. In the present paper we prove Σ≥ √π/2 ≈ 0.886, by this means improving a result of Karatsuba. For the proof we use a quantitative two-dimensional version of the central limit theorem for lacunary trigonometric series, which in its original form is due to Salem and Zygmund.

Related

• On the L¹ norm of an exponential sum involving the divisor function
• L¹ means of exponential sums with multiplicative coefficients. I
• Linnik's large sieve and the L¹ norm of exponential sums
• On the mean value of the magnitude of an exponential sum involving the divisor function
• Moments of an exponential sum related to the divisor function