 Closure bounds for small skew-sumset scalars

Write $D(t)=|A+tA|/|A|$ and suppose some nonzero $b$ has $D(b)\le K$. The mixed-set Plünnecke-Ruzsa inequality gives $|3A-2A|\le K^5|A|$ and $|A+A|\le K^2|A|$. 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 $D(-r)\le K^{2}D(r)$. These bounds transfer controlled additive expansion through a bounded-complexity expression in scalars.