A proof of Erdős's $B+B+t$ conjecture

Abstract

We show that every set A of natural numbers with positive upper density can be shifted to contain the restricted sumset

Sequence entropy tuples and mean sensitive tuples

“…Our results provide more information in the general case, and the proofs work on the classical definition of sequence entropy pairs introduced in [15]. It is worth noting that the proofs depend on a new interesting ergodic measure decomposition result (Lemma 4.3), which was applied to prove the profound Erdös's conjectures in the number theory by Kra et al [20,21]. This decomposition may have more applications because it has the hybrid topological and Borel structures.…”

Infinite partial sumsets in the primes

“…of positive upper density, there existed an infinite subset B of A and a natural number t such that b + b ′ + t ∈ A for all distinct b, b ′ in B. This conjecture was recently proven in [18], using techniques from ergodic theory and topological dynamics. It is natural to ask whether the same result holds if the set A is replaced by the primes P = {2, 3, 5, .…”

“…Remark 2.2. One can view the above construction using the dynamical systems framework of [18] (and indeed our arguments were initially inspired by this framework). Let X denote 4 the orbit closure of the primes P, that is to say the closure in the product topology of the shifts P + t, t ∈ Z of the primes, viewed as elements of the Cantor space 2 Z = {A : A ⊂ Z}.…”

“…Remark 1.4. In view of the results in [18], one might also ask if Theorem 1.3 continued to hold if one replaced the set of primes P by some positive (relative) density subset. However, this is not the case.…”

Sequence entropy tuples and mean sensitive tuples

“…Their proof relies on the deep equivalent characterization of measurable sequence entropy pairs developed by Kerr and Li [19] using the combinatorial notion of independence. Our results provide more information in general case, and proofs work on the classical definition of sequence entropy pairs introduced in [15] and depend on a new interesting ergodic measure decomposition result (Lemma 4.3), which was applied to prove the profound Erdös's conjectures in the number theory by Kra, Moreira, Richter and Robertson [20,21].…”