On an uncertainty principle for small index subgroups of finite fields

In this article, we continue the study of the nonvanishing minors property initiated by Garcia, Karaali, and Katz, for the compressed Fourier matrix attached to a subgroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>H</m:mi> </m:math> H of the multiplicative group of a finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">F</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> {{F}}q and a character <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>χ</m:mi> </m:math> χ defined over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>H</m:mi> </m:math> H . Here, we provide a characterization of this aforementioned property for symmetries arising from an index-3 subgroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>H</m:mi> </m:math> H and a nontrivial character <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>χ</m:mi> </m:math> χ .