= Multiplicative energy sumset bound
{title2=$E(A,B)\leq4\lceil\log|B|\rceil|A+A||B+B|$}
For finite sets of positive real numbers $A,B$ with $|B|\geq2$, $E(A,B)\leq4\lceil\log|B|\rceil|A+A||B+B|$, with natural or binary <logarithm>. Partition ray occupancies of $A\times B$ into $\lceil\log|B|\rceil$ classes of multiplicative width $e$, or width $2$ for the binary convention. <Injectivity of sums on two distinct rays> and disjoint <open planar sectors> bound each class's sum of squared occupancies by $4|A+A||B+B|$. A singleton class contributes at most $|A||B|$. Combining this with the energy lower bound gives a sum-product inequality.
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