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• W4313591981 abstract "Fix <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha element-of left-parenthesis 0 comma 1 slash 3 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha in (0,1/3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that, from a topological point of view, almost all sets <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A subset-of-or-equal-to double-struck upper N> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>Asubseteq mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have the property that, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A prime equals upper A> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mi class=MJX-variant mathvariant=normal>′<!-- ′ --></mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>A^prime =A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all but <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=o left-parenthesis n Superscript alpha Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>o(n^{alpha })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A prime> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mi class=MJX-variant mathvariant=normal>′<!-- ′ --></mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>A^prime</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not a nontrivial sumset <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper B plus upper C> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>B+C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, almost all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are totally irreducible. In addition, we prove that the measure analogue holds with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha equals 1> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha =1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W4313591981 created "2023-01-06" @default.
• W4313591981 creator A5008510143 @default.
• W4313591981 date "2023-06-06" @default.
• W4313591981 modified "2023-09-26" @default.
• W4313591981 title "Almost all sets of nonnegative integers and their small perturbations are not sumsets" @default.
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• W4313591981 doi "https://doi.org/10.1090/proc/16392" @default.
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