Heisenberg uniqueness pairs for a certain system of irregularly distributed parallel lines associated with prime

In this article, we establish necessary and sufficient conditions for Heisenberg uniqueness pairs corresponding to a specific system of parallel lines that are irregularly distributed and associated with primes greater than or equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Additionally, inspired by the work of D. B. Babot [Proc. Amer. Math. Soc. 141 (2013), pp. 3899–3904], we present some noteworthy examples of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda subset-of double-struck upper R times left-bracket 0 comma 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda \subset \mathbb R\times [0,2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which correspond to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R times StartSet 0 comma 1 comma p EndSet"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo> × </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb R\times \{0,1,p\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 3 comma 4 comma 5"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p=3,4,5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , along with a system of finitely many consecutive parallel lines in the plane.