Erdős Problem #3: sets with divergent reciprocal sum contain arbitrarily long arithmetic progressions

Theorem 1.1 claims, for each fixed $k\ge3$, $r_k(N)\le C_kN\exp(-c_k(\log N)^{\varepsilon_k})$ with unspecified positive constants depending on $k$; equivalently a density-$\alpha$ subset of $[N]$ has a $k$-term progression once $\log N\ge A_k(2+\log(1/\alpha))^{A_k}$. Summing over dyadic blocks gives Erdos's conjecture, and even the stronger criterion with weights $(\log(2+a))^B/a$, and recovers Green-Tao for dense subsets of the primes. The method is a density increment on triangular polynomial cells with controlled precision. No improvement of the three-term exponent is claimed, the exponents $\varepsilon_k$ are not optimized, and it does not give an asymptotic formula for $r_k(N)$ (Erdos Problem #142).

Theorem 1.1 claims, for each fixed $k\ge3$, $r_k(N)\le C_kN\exp(-c_k(\log N)^{\varepsilon_k})$ with unspecified positive constants depending on $k$; equivalently a density-$\alpha$ subset of $[N]$ has a $k$-term progression once $\log N\ge A_k(2+\log(1/\alpha))^{A_k}$. Summing over dyadic blocks gives Erdos's conjecture, and even the stronger criterion with weights $(\log(2+a))^B/a$, and recovers Green-Tao for dense subsets of the primes. The method is a density increment on triangular polynomial cells with controlled precision. No improvement of the three-term exponent is claimed, the exponents $\varepsilon_k$ are not optimized, and it does not give an asymptotic formula for $r_k(N)$ (Erdos Problem #142).