A new type of deterministic Salem sets and its spectrality
Chun-Kit Lai, Ruxi Shi, Yu-Hao Xie
math.CA
Sep 19, 2026 · v1
math.FA
TL;DR
The main theorem (spectrality of the Cantor–Moran measure and the Salem property of the Moran set) is formalized in Lean 4 with Mathlib and kernel-checked.
Abstract
For all $0< s \le 1$, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to $s$. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis $\{e^{2πi λx}: λ\in Λ\}$ for its $L^2$ space. This gives the first examples of singular Salem spectral measures in ${\mathbb R}^1$.
Problem
Deterministic Salem sets of every dimension s in (0,1] are sought, along with singular spectral measures on R that also have optimal Fourier decay. No singular Salem spectral measures in R^1 were previously known.
Approach
Homogeneous Cantor–Moran sets are built with contraction ratios that are reciprocals of prime powers. The digit sets come from coordinate functions of powers in finite fields F_{p^r}. The Weil bound controls how far the mask functions are from Dirichlet kernels, which gives fast Fourier decay. Spectrality is proved using unitary Hadamard-type matrices. The main conclusions were formalized in Lean 4 using Mathlib.
Results
For every 0<s≤1, the constructed Moran set has Fourier and Hausdorff dimension both equal to s. Its equal-weighted Cantor–Moran measure is spectral, giving the first singular Salem spectral measures in R^1. The proofs of these conclusions were checked by the Lean kernel.
