A tight structure theorem for sumsets

<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals left-brace 0 equals a 0 greater-than a 1 greater-than midline-horizontal-ellipsis greater-than a Subscript script l plus 1 Baseline equals b right-brace"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>></mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>></mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A = \{0 = a_0 > a_1 > \cdots > a_{\ell + 1} = b\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite set of non-negative integers. We prove that the sumset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N upper A"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">NA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a certain easily-described structure, provided that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-slanted-equals b minus script l"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>⩾<!-- ⩾ --></mml:mo> <mml:mi>b</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N \geqslant b-\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, as recently conjectured (see A. Granville and G. Shakan [Acta Math. Hungar. 161 (2020), pp. 700–718]). We also classify those sets <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> for which this bound cannot be improved.</p>