Jakub Konieczny

Weakly mixing sets of integers and polynomial equations

We investigate polynomial patterns which can be guaranteed to appear in \emph{weakly mixing} sets introduced by introduced by Furstenberg and studied by Fish. In particular, we prove that if

Sets of recurrence as bases for the positive integers

A \subset \mathbb N

Combinatorial properties of Nil-Bohr sets

is a weakly mixing set and

Gowers norms for the Thue-Morse and Rudin-Shapiro sequences.

p(x) \in \mathbb Z[x]

Sparse generalised polynomials

a polynomial of odd degree with positive leading coefficient, then all sufficiently large integers

A density version of Cobham's theorem.

N

Automatic sequences and generalised polynomials.

can be represented as

Arcwise connectedness of the set of ergodic measures of hereditary shifts

N = n_1 + n_2

On uniformity of

, where

q

p(n_1) + m,\ p(n_2) + m \in A