arXiv subjects

Xinggang He

Publications and source records attributed to Xinggang He.

3 recordsLinked to original sources

arXiv2022

Classification of spectral self-similar measures with four-digit elements

Let $μ$ be a self-similar measure generated by iterated function system of four maps of equal contraction ratio $0<ρ<1$. We study when $μ$ is a spectral measure which means that it admits an exponential orthonormal basis $\{e^{2πi λx}\}_{λ\inΛ}$ in $L^2(μ)$. By combining previous results of many authors and a careful study of some new cases, we completely classify all spectral self-similar measures with four maps. Moreover, the case allows us to propose a modified Łaba-Wang conjecture concerning when the self-similar measures are spectral in general cases.

math.CA↗

arXiv2013

On the range of $\sum_{n=1}^\infty\pm c_n$

Let $\{c_n\}_{n=1}^\infty$ be a sequence of complex numbers. In this paper we answer when the range of $\sum_{n=1}^\infty\pm c_n$ is dense or equal to the complex plane. Some examples are given to explain our results. As its application, we calculate the Hausdorff dimension of the level sets of a Rademacher series with complex coefficients.

math.FA↗

arXiv2013

Spectrality of Self-Similar Tiles

We call a set $K \subset {\mathbb R}^s$ with positive Lebesgue measure a {\it spectral set} if $L^2(K)$ admits an exponential orthonormal basis. It was conjectured that $K$ is a spectral set if and only if $K$ is a tile (Fuglede's conjecture). Despite the conjecture was proved to be false on ${\mathbb R}^s$, $s\geq 3$ ([T], [KM2]), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties for the class of self-similar tiles $K$ in ${\mathbb R}$ that has a product structure on the associated digit sets. We show that any strict product-form tiles and the associated modulo product-form tiles are spectral sets. As for the converse question, we give a pilot study for the self-similar set $K$ generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality, and verify Fuglede's conjecture for this special case.

math.FA↗