2016/07/18 by Misha Rudnev, Rudnev, Misha, Ilya D. Shkredov +3 · 4 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #math.CO #math.NT #msc:11B75 #msc:68R05
paper · pdf · doi:10.4171/rmi/1126
25 pages. A new best sum-product exponent is attained over the reals via a new corollary which locally improves the two recent sum-product papers by Konyagin and Shkredov
arxiv created 2017/06/05 · arxiv updated 2017/06/06 · openalex publication_date 2019/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove new exponents for the energy version of the Erdős-Szemerédi 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 (4)/(3)+(1)/(1509). An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.