The size of subset of {1,…,N} without distinct degree polynomial progressions is O(N/(log log N)^c)

You need to know: Polynomial, degree of a polynomial, notation ${\mathbb Z}[y]$ for the set of polynomials in 1 variable with integer coefficients, notation $[N]$ for the set $\{1,2,\dots,N\}$ .

Background: Let $P=(P_1, \dots, P_m)$ be a finite set of polynomials $P_i \in {\mathbb Z}[y]$ . For integer $N>0$ , let $r_P(N)$ be the size of the largest subset of $[N]$ containing no subset of the form $x, x+P_1(y), \dots, x+P_m(y)$ with $y\neq 0$ .

The Theorem: On 1st September 2019, Sarah Peluse submitted to arxiv a paper in which she proved that if all polynomials $P_i$ in $P$ have distinct degrees and zero constant terms, then there exists a constant $c$ depending on $P_1, \dots, P_m$ such that
$r_P(N) \leq \frac{N}{(\log\log N)^c}$ .

Short context: Let $r_k(N)$ be the cardinality of the largest subset of $[N]$ which contains no nontrivial k-term arithmetic progressions. Famous Szemerédi’s theorem states that $\lim\limits_{N\to\infty}\frac{r_k(N)}{N}=0$ . In 2000, Gowers proved an explicit bound $r_k(N) < N(\log\log N)^{-c}$ for some constant $c=c(k)>0$ . In 1996, Bergelson and Leibman extended Szemerédi’s theorem to polynomial progressions and proved that $\lim\limits_{N\to\infty} \frac{r_P(N)}{N} = 0$ , if polynomials $P_i$ all have zero constant terms. The Theorem establishes an explicit upper bound on $r_P(N)$ , provided that all polynomials $P_i$ have distinct degrees.