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$ABC$ sum-product theorems for Katz-Tao sets

Tuomas Orponen

TL;DR

The paper addresses the $ABC$ sum-product problem for $oldsymbol{ ext{$ au$-separated}}$ sets in $[0,1]$ under Katz–Tao spacing. It introduces two sharp variants that remove the requirement that $|B|$ and $|C|$ match their non-concentration exponents, one using a Frostman condition on $C$ to compensate for a weaker hypothesis on $B$. The approach combines a new auxiliary ABC proposition with a careful multi-scale, uniform-set analysis and standard additive-combinatorics tools to produce nontrivial expansions $|igl\{a+cb:(a,b)igrigoldsymbol{)}|_{ ho}$ for some $c earrow C$, and to deduce the classical ABC conjecture from these results. The findings connect δ-discretized sum-product phenomena with Frostman/Katz–Tao structure, offering sharp, scale-aware bounds and improving the understanding of how non-concentration interacts with expansion in sum–product problems.

Abstract

I prove two variants of the $ABC$ sum-product theorem for $δ$-separated sets $A,B,C \subset [0,1]$ satisfying Katz-Tao spacing conditions. The main novelty is that the cardinality of the sets $B,C$ need not match their non-concentration exponent. The new $ABC$ theorems are sharp under their respective hypotheses, and imply the previous one.

$ABC$ sum-product theorems for Katz-Tao sets

TL;DR

The paper addresses the sum-product problem for au sets in under Katz–Tao spacing. It introduces two sharp variants that remove the requirement that and match their non-concentration exponents, one using a Frostman condition on to compensate for a weaker hypothesis on . The approach combines a new auxiliary ABC proposition with a careful multi-scale, uniform-set analysis and standard additive-combinatorics tools to produce nontrivial expansions for some , and to deduce the classical ABC conjecture from these results. The findings connect δ-discretized sum-product phenomena with Frostman/Katz–Tao structure, offering sharp, scale-aware bounds and improving the understanding of how non-concentration interacts with expansion in sum–product problems.

Abstract

I prove two variants of the sum-product theorem for -separated sets satisfying Katz-Tao spacing conditions. The main novelty is that the cardinality of the sets need not match their non-concentration exponent. The new theorems are sharp under their respective hypotheses, and imply the previous one.

Paper Structure

This paper contains 12 sections, 17 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Proof and paper outline
  4. Preliminaries
  5. Frostman and Katz-Tao sets
  6. Uniform sets and branching functions
  7. Background on Lipschitz functions
  8. Lemmas from additive combinatorics
  9. An auxiliary proposition
  10. Proof of Theorem \ref{['c3']}
  11. Proof of Theorem \ref{['c4']}
  12. Deducing Theorem \ref{['thm:ABCConjecture']} from Theorem \ref{['c3']}

Key Result

Theorem 1.1

Let $\alpha \in [0,1)$ and $\beta,\gamma > 0$ satisfy $\beta + \gamma > \alpha$. Then, there exist $\chi,\delta_{0} \in (0,\tfrac{1}{2}]$ such that the following holds for all $\delta \in 2^{-\mathbb{N}} \cap (0,\delta_{0}]$. Let $A,B,C \subset \delta \mathbb{Z} \cap [0,1]$ be sets satisfying: Then there exists $c \in C$ such that In particular $\max_{c \in C} |A + cB|_{\delta} \geq \delta^{-\ch

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3: Frostman $(\delta,s,C)$-set
  • Remark 1.4
  • Definition 1.5: Katz-Tao $(\delta,s,C)$-set
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 37 more