The Sum-Product conjecture is false for real numbers
11 hours ago
- #sum-product conjecture
- #real numbers
- #number theory
- The sum-product conjecture is disproved for real numbers.
- An arbitrarily large set A of algebraic integers is constructed such that max(|A+A|, |AA|) ≤ |A|^{2-c} with c>0.
- The many sums and products conjecture is disproved for any k≥3, showing max(|kA|, |A^{(k)}|) ≤ |A|^{C log k/log log k}.
- Similar constructions are extended to p-adics, finite fields, and function fields.
- New lower bounds for solutions to linear equations in multiplicative groups and unit equations are obtained.