Tilings, sub-tilings, and spectral sets on <i>p</i> -adic space

Abstract In this article, we provide a characterization of tilings, sub-tilings, and spectral sets on the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -adic space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msubsup> </m:math> {{\mathbb{Q}}}_{p}^{d} . Our methods are based on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -adic distributions and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -adic Fourier analysis. We also obtained a sufficient condition and a necessary condition on tiling of functions by translation on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msubsup> </m:math> {{\mathbb{Q}}}_{p}^{d} .