On The Energy Variant of the Sum-Product Conjecture

We prove new exponents for the energy version of the Erd\H{o}s-Szemer\'edi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the "few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to $\frac{4}{3}+\frac{1}{1509}$. An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.