Prime Power Residues and Blocking Sets

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> be a fixed odd prime. We show that a finite subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> of integers, not containing any perfect <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:msup> </mml:math> power, contains a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:msup> </mml:math> power modulo almost every prime if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> corresponds to a blocking set (with respect to hyperplanes) in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">PG</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>𝔽</mml:mi> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> is the number of distinct prime divisors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -free parts of elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> . As a consequence, the property of a subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> to contain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:msup> </mml:math> power modulo almost every prime <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> is invariant under geometric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> -equivalence defined by an element of the projective general linear group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">PGL</mml:mi> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>𝔽</mml:mi> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Employing this connection between two disparate branches of mathematics, Galois geometry and number theory, we classify, and provide bounds on the sizes of, minimal such sets <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>B</mml:mi> </mml:math> .