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Discretized sum-product type problems: Energy variants and Applications

Quy Pham, Thang Pham, Chun-Yen Shen

TL;DR

Non-trivial estimates for the additive discretized energy ofsum c in C that depend on non-concentration conditions of the sets are proved.

Abstract

In this paper, we provide estimates for the additive discretized energy of \[\sum_{c\in C} |\{(a_1, a_2, b_1, b_2)\in A^2\times B^2: |(a_1 +cb_1) - (a_2 + cb_2)|\le δ\}|_δ,\] that depend on non-concentration conditions of the sets. Our proof follows the Guth-Katz-Zahl approach (2021) with appropriate changes along the way clarifying and optimizing many of the steps. Several applications will also be discussed.

Discretized sum-product type problems: Energy variants and Applications

TL;DR

Non-trivial estimates for the additive discretized energy ofsum c in C that depend on non-concentration conditions of the sets are proved.

Abstract

In this paper, we provide estimates for the additive discretized energy of that depend on non-concentration conditions of the sets. Our proof follows the Guth-Katz-Zahl approach (2021) with appropriate changes along the way clarifying and optimizing many of the steps. Several applications will also be discussed.
Paper Structure (13 sections, 25 theorems, 170 equations)

This paper contains 13 sections, 25 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications on the $A+cB$ problem
  4. An energy theorem via incidence bounds
  5. Basic lemmas from Additive Combinatorics
  6. A structural theorem: good subsets of $B$ and $C$
  7. Proof of Theorem \ref{['thm_energy_III']}
  8. Dense case
  9. Gap case
  10. Concluding the proof
  11. Proof of Theorem \ref{['thm-G1l']} and Theorem \ref{['thm-G2']}
  12. Proof of Theorem \ref{['thm_energy_II']} and Theorem \ref{['a+cba41']}
  13. Acknowledgements

Key Result

Theorem 1.1

Given $\alpha \in (0,1)$ and $\gamma,\eta > 0$, there exist $\epsilon_{0},\epsilon > 0$, depending only on $\alpha, \gamma, \eta$, such that the following holds for all sufficiently small $\delta > 0$. Let $\nu$ be a probability measure on $[0,1]$ satisfying $\nu(B(x,r)) \le r^{\gamma}$ for all $x \ Here $|\cdot|_{\delta}$ refers to the $\delta$-covering number of $A$, namely the size of the small

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 24 more