Rational points in Cantor sets and spectral eigenvalue problem for self-similar spectral measures

Abstract Given <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>ℕ</m:mi> <m:mrow> <m:mi/> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> {q\in\mathbb{N}_{\geq 3}} and a finite set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>⊂</m:mo> <m:mi>ℚ</m:mi> </m:mrow> </m:math> {A\subset\mathbb{Q}} , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo maxsize="260%" minsize="260%">{</m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:munderover> <m:mfrac> <m:msub> <m:mi>a</m:mi> <m:mi>i</m:mi> </m:msub> <m:msup> <m:mi>q</m:mi> <m:mi>i</m:mi> </m:msup> </m:mfrac> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mtext> for all </m:mtext> <m:mo>⁢</m:mo> <m:mi>i</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> </m:mrow> <m:mo maxsize="260%" minsize="260%">}</m:mo> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> K(q,A)=\Bigg{\{}\sum_{i=1}^{\infty}\frac{a_{i}}{q^{i}}:a_{i}\in A\text{ for % all }i\in\mathbb{N}\Bigg{\}}. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>ℕ</m:mi> <m:mrow> <m:mi/> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> {p\in\mathbb{N}_{\geq 2}} let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>D</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> {D_{p}\subset\mathbb{R}} be the set of all rational numbers having a finite p -ary expansion. We show in this paper that for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>ℕ</m:mi> <m:mrow> <m:mi/> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> {p\in\mathbb{N}_{\geq 2}} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>gcd</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {\gcd(p,q)=1} , the intersection <jats:inline-formula id="j_foru