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• W2007694992 abstract "We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper E> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>E</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a subset of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional vector space over a finite field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper F Subscript q> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than-or-equal-to 3> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) of cardinality <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue script upper E EndAbsoluteValue greater-than-or-equal-to left-parenthesis d minus 1 right-parenthesis q Superscript d minus 1> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>E</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>|mathcal {E}| geq (d-1)q^{d - 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the set of volumes of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional parallelepipeds determined by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper E> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>E</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> covers <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper F Subscript q> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This bound is sharp up to a factor of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis d minus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(d-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as taking <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper E> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>E</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {E}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis d minus 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(d - 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-hyperplane through the origin shows." @default.
• W2007694992 created "2016-06-24" @default.
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• W2007694992 date "2013-06-04" @default.
• W2007694992 modified "2023-09-24" @default.
• W2007694992 title "On the volume set of point sets in vector spaces over finite fields" @default.
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• W2007694992 doi "https://doi.org/10.1090/s0002-9939-2013-11630-8" @default.
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