= Closure bounds for small skew-sumset scalars
{title2=$D(r+s)\leq K^5D(r)D(s),\quad D(rs)\leq K^5D(r)^2D(s)$}
Write $D(t)=|A+tA|/|A|$ and suppose some nonzero $b$ has $D(b)\leq K$. The mixed-set <Plünnecke-Ruzsa inequality> gives $|3A-2A|\leq K^5|A|$ and $|A+A|\leq K^2|A|$. The <Ruzsa covering lemma> covers $tA$ by at most $D(t)$ translates of $A-A$. Covering two dilates proves the addition bound. First covering $sA$, then both copies of $rA$ in $r(A-A)$, proves the multiplication bound. The <Ruzsa triangle inequality> also gives $D(-r)\leq K^2D(r)$. These bounds transfer controlled additive expansion through a bounded-complexity expression in <scalars>.
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