Let P 1 , . . . , P m ∈ Z[y] be any linearly independent polynomials with zero constant term. We show that there exists γ > 0 such that any subset of F q of size at least q 1−γ contains a nontrivial polynomial progression x, x + P 1 (y), . . . , x + P m (y), provided the characteristic of F q is large enough.

We show that any subset of [N ] of density at least (log log N ) −2 −157 contains a nontrivial progression of the form x, x + y, x + y 2 . This is the first quantitatively effective version of the Bergelson-Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. In the course of the proof, we also develop a quantitative version of a special case of a concatenation theorem of Tao and Ziegler, with polynomial bounds.

We show that sets of integers lacking the configuration $x$, $x+y$, $x+y^2$ have at most polylogarithmic density.

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Bourgain and Chang recently showed that any subset of Fp of density ≫ p −1/15 contains a nontrivial progression x, x + y, x + y 2 . We answer a question of theirs by proving that if P1, P2 ∈ Z[y] are linearly independent and satisfy P1(0) = P2(0) = 0, then any subset of Fp of density ≫P 1 ,P 2 p −1/24 contains a nontrivial polynomial progression x, x + P1(y), x + P2(y).

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers α so that either (α, α 2 ) or (α, α − α 2 ) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u, u ′ ) with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u, u ′ ) has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

Let $P_1,\dots ,P_m\in \mathbb{Z} [y]$ be polynomials with distinct degrees, each having zero constant term. We show that any subset A of $\{1,\dots ,N\}$ with no nontrivial progressions of the form $x,x+P_1(y),\dots ,x+P_m(y)$ has size $|A|\ll N/(\log \log {N})^{c_{P_1,\dots ,P_m}}$ . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

We show that almost every entry in the character table of S N {S_{N}} is divisible by any fixed prime as N → ∞ {N\to\infty} . This proves a conjecture of Miller.