文摘
We study spectral properties of the self-affine measure class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si116.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=c3b578b77d2661841fe0bebf5a29745f" title="Click to view the MathML source">μM,Dclass="mathContainer hidden">class="mathCode"> generated by an expanding integer matrix class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si264.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=896df5673b3e68e825c1b43849429d32" title="Click to view the MathML source">M∈Mn(Z)class="mathContainer hidden">class="mathCode"> and a consecutive collinear digit set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si131.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=dfa9cb06d4a31d6498b89f37c47fb1d3" title="Click to view the MathML source">D={0,1,…,q−1}vclass="mathContainer hidden">class="mathCode"> where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si115.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=bfef620d8742554af638bd68a685a048" title="Click to view the MathML source">v∈Zn∖{0}class="mathContainer hidden">class="mathCode"> and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si52.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=6cef4677b5a4a6cc407922e007352eff" title="Click to view the MathML source">q≥2class="mathContainer hidden">class="mathCode"> is an integer. Some sufficient conditions for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si116.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=c3b578b77d2661841fe0bebf5a29745f" title="Click to view the MathML source">μM,Dclass="mathContainer hidden">class="mathCode"> to be a spectral measure or to have infinitely many orthogonal exponentials are given. Moreover, for some special cases, we can obtain a necessary and sufficient condition on the spectrality of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303159&_mathId=si116.gif&_user=111111111&_pii=S0022123616303159&_rdoc=1&_issn=00221236&md5=c3b578b77d2661841fe0bebf5a29745f" title="Click to view the MathML source">μM,Dclass="mathContainer hidden">class="mathCode">. Our study generalizes the one dimensional results proved by Dai, et al. and .