Hyperplane covers of finite spaces and applications

Given a prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript p Baseline left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_p(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the minimal number of hyperplanes in an irredundant covering of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript p Superscript n"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathbb {F}_p^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the normal vectors of the hyperlanes span the whole space. The function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript p Baseline left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_p(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> appears to be in connection to several longstanding conjectures in linear algebra and group theory, such as the Alon-Jaeger-Tarsi conjecture, the Additive Basis conjecture, and a conjecture of Pyber on irredundant coset covers of abelian groups. We prove that <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript p Baseline left-parenthesis n right-parenthesis equals normal upper Omega left-parenthesis StartFraction log p Over log log p EndFraction dot n right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo> ⋅ </mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} f_p(n)=\Omega \left (\frac {\log p}{\log \log p}\cdot n\right ), \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> and use this result to make substantial progress on each of the aforementioned conjectures.