Pseudorandomness and Diffraction
Abstract
Pseudorandom structures are structures that behave like random ones, without necessarily being random themselves. It is a prominent, and evidently very difficult, conjecture that V(n) = λcos ( 2 π( x1 + n x2 + (n(n−1))/2 α) ) is pseudorandom from a Schrödinger operator perspective in that the Schrödinger operator in ℓ2(ℤ) with potential V displays Anderson localization, that is, pure point spectrum with exponentially decaying eigenfunctions for almost all parameter values --- the same spectral features as those produced by random potentials. We show that V is pseudorandom in terms of its diffraction properties, that is, the associated diffraction measure is purely absolutely continuous for all λnot= 0, all irrational α, and all x1,x2 --- which is the case as well for the random case. Our result gives further evidence for the conjecture in the Schrödinger case and it elucidates the apparent dual behavior of Schrödinger spectral measures and diffraction measures.
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BibTeX
@misc{damanik2026pseudorandomness,
title = {{Pseudorandomness and Diffraction}},
author = {David Damanik and Nicolae Strungaru},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28917},
eprint = {2608.28917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28917},
}
RIS
TY - UNPB
TI - Pseudorandomness and Diffraction
AU - David Damanik
AU - Nicolae Strungaru
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28917
UR - https://arxiv.org/abs/2608.28917
AB - Pseudorandom structures are structures that behave like random ones, without necessarily being random themselves. It is a prominent, and evidently very difficult, conjecture that V(n) = λcos ( 2 π( x_1 + n x_2 + (n(n−1))/2 α) ) is pseudorandom from a Schrödinger operator perspective in that the Schrödinger operator in ℓ^2(ℤ) with potential V displays Anderson localization, that is, pure point spectrum with exponentially decaying eigenfunctions for almost all parameter values --- the same spectral features as those produced by random potentials. We show that V is pseudorandom in terms of its diffraction properties, that is, the associated diffraction measure is purely absolutely continuous for all λnot= 0, all irrational α, and all x_1,x_2 --- which is the case as well for the random case. Our result gives further evidence for the conjecture in the Schrödinger case and it elucidates the apparent dual behavior of Schrödinger spectral measures and diffraction measures.
ER -