Positive Lower Natural Density Does Not Ensure Carleson Frame Sampling

Abstract

We construct a locally finite, strictly increasing real sampling sequence with lower natural density 1/3 and upper natural density 4/3 for which a sampled Carleson orbit is Bessel but has no positive frame lower bound. The diagonal spectrum is fixed explicitly as z_j = exp(-2^(-j)). At each time 4^k, the sequence places 4^k distinct samples in an interval of length less than 1/2. Finite-dimensional cancellation at the cluster centres and an explicit Hilbert–Schmidt estimate for the within-cluster error yield finitely supported unit vectors whose energy over the entire sampling sequence tends to zero. This gives a negative answer to the weaker universal sufficiency statement discussed in Remark 4.2 of Krishtal and Miller, Block Diagonal Carleson Frames. For comparison, an appendix proves a uniform lower bound for an integer sampling family on the same spectrum. This deposit contains the eight-page preprint and its complete LaTeX source. The manuscript includes an AI-assistance disclosure.