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- W4283701082 abstract "Abstract A Borel probability measure μ on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>ℝ</m:mi> </m:math> {mathbb{R}} is called a spectral measure if it has an exponential orthonormal basis for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{2}(mu)} . In this paper, we consider the spectrality of the self-similar measure <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>μ</m:mi> <m:mrow> <m:mi>ρ</m:mi> <m:mo>,</m:mo> <m:mi>D</m:mi> </m:mrow> </m:msub> </m:math> {mu_{rho,D}} generated by <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>ρ</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<rho<1} and the product-form digit set <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>D</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:msub> <m:mi>p</m:mi> <m:mn>0</m:mn> </m:msub> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>L</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=normal>…</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> <m:mo>⊕</m:mo> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:msub> <m:mi>p</m:mi> <m:mn>1</m:mn> </m:msub> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=normal>…</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> <m:mo>⊕</m:mo> <m:mi mathvariant=normal>⋯</m:mi> <m:mo>⊕</m:mo> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:msub> <m:mi>p</m:mi> <m:mi>s</m:mi> </m:msub> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>L</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=normal>…</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=false>}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> D=N^{p_{0}}L_{0}{0,1,ldots,N-1}oplus N^{p_{1}}L_{1}{0,1,ldots,N-1}% opluscdotsoplus N^{p_{s}}L_{s}{0,1,ldots,N-1}, where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {Ngeq 2} is a prime number, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:msub> <m:mi>p</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo><</m:mo> <m:msub> <m:mi>p</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo><</m:mo> <m:mi mathvariant=normal>⋯</m:mi> <m:mo><</m:mo> <m:msub> <m:mi>p</m:mi> <m:mi>s</m:mi> </m:msub> </m:mrow> </m:math> {0leq p_{0}<p_{1}<cdots<p_{s}} and <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:msub> <m:mi>L</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>,</m:mo> <m:mi>N</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(L_{i},N)=1} for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>s</m:mi> </m:mrow> </m:math> {0leq ileq s} . We show that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi>μ</m:mi> <m:mrow> <m:mi>ρ</m:mi> <m:mo>,</m:mo> <m:mi>D</m:mi> </m:mrow> </m:msub> </m:math> {mu_{rho,D}} is a spectral measure if and only if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>ρ</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>N</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo></m:mo> <m:mi>M</m:mi> </m:mrow> </m:mrow> </m:math> {rho^{-1}=N^{q}M} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>≠</m:mo> <m:mrow> <m:mpadded width=+3.3pt> <m:msub> <m:mi>p</m:mi> <m:mi>j</m:mi> </m:msub> </m:mpadded> <m:mspace width=veryverythickmathspace /> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mi /> <m:mo lspace=2.5pt rspace=5.8pt>mod</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {p_{i}neq p_{j}~{}(bmod~{}q)} for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≠</m:mo> <m:mi>j</m:mi> <m:mo>≤</m:mo> <m:mi>s</m:mi> </m:mrow> </m:math> {0leq ineq jleq s} , where <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>M</m:mi> <m:mo>,</m:mo> <m:mi>N</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(M,N)=1} ." @default.
- W4283701082 created "2022-06-30" @default.
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- W4283701082 date "2022-06-29" @default.
- W4283701082 modified "2023-10-16" @default.
- W4283701082 title "Spectrality of self-similar measures with product-form digits" @default.
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