Thue-Morse准晶的光谱和稀疏现象
On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon
摘要：
本文研究了Thue-Morse准晶上加权Dirac梳的谱，对于Bragg峰，用Berryeri-Taylor猜想，对于奇异连续分量，用Aubry-Godrèche-Luck猜想刻画了谱的零测度集.本文给出了加权狄拉克梳的傅里叶变换的回火分布分解。我们证明了Thue-Morse序列的$p$ -稀疏和的渐近算法（Dumont; Goldstein，Kelly and Speer; Grabner; Drmota and Skalba，.），即数字和函数的分形性，在描述谱的奇异连续部分中起着重要作用，结合Peyrière和M.奎菲莱克在一部分奇异分量上的逼近测度序列的优势标度由某些不等式控制，其中涉及实二次域的类数和调节器.
英文摘要：
The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyrière and M. Queffélec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.
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