Second-order lacunary a.e. Fourier convergence in ℝ^n

Determine whether, for 2 ≤ p < 2n/(n−1) and f ∈ L^p(ℝ^n), the spherical partial sums S_{2^k(1+2^{-j})}f(x) converge almost everywhere to f(x) along the second-order lacunary sequence {2^k(1+2^{-j})}_{k∈ℤ, j∈ℕ}.

Background

The paper proves a.e. convergence for the lacunary sequence {2^k} in the p-range 2 ≤ p < 2n/(n−1). It also provides positive results for second-order lacunary sequences under stronger logarithmic integrability conditions on the Fourier transform.

However, in the general L^p setting without extra smoothness, it is unknown whether a.e. convergence holds along second-order lacunary sequences, reflecting a higher-dimensional analogue of maximal Hilbert transform phenomena in one dimension.