 Solymosi sum-product theorem over the complex numbers

ID: solymosi-sum-product-theorem-over-the-complex-numbers

For finite

U, V, W \subseteq C

with

U \neq = {0}

and

W \neq = {0}

∣ U + V ∣ ∣ U W ∣ \geq \frac{1}{56} ∣ U ∣^{3/2} ∣ V ∣^{1/2} ∣ W ∣^{1/2} .

In particular, addition and multiplication cannot both expand a nontrivial finite set of complex numbers only slightly.

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