On the energy variant of the sum-product conjecture

By M. Rudnev, Ilya D. Shkredov, Sophie Stevens. Revista matemática iberoamericana, 2016

51 citations recorded by the source corpus.

We prove new exponents for the energy version of the Erdős-Szemeredi 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/497fccae4df31e1d584ae34dc6b993bcb6e738f4

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In Algebra and Number Theory

1. Ilya D. Shkredov
2. Tom Sanders
3. Ben Joseph Green

In Discrete Mathematics and Combinatorics

1. Ilya D. Shkredov
2. M. Rudnev
3. Sophie Stevens
4. Tom Sanders
5. Ben Joseph Green

Citations in the corpus

• On asymptotic formulae in some sum–product questions (2018, background): “, AB := {ab : a ∈ A, b ∈ B}, and A/B := {a/b : a ∈ A, b ∈ B, b 6= 0 }, correspondingly. Our paper is devoted to so–called the sum–product phenomenon in Fp which was developed in papers [1]–[17], [27]—[43], [48]—[53], [57], [58] and in many others. This is extensively growing area of mathematics with”