Fix $k,s\in\mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ of degree $k$ and with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$. This provides a density strengthening...
We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\geq 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\ldots, x+ky,xy\}$...
Fix $s\in\mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in\mathbb{Z}[y]$ with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),xy^s\}$ for arbitrarily large $x>y\geq 2$. When $s=1$, this strengthens Moreira's partition-regulari...
F. Richter, Konstantinos Tsinas· 0 citations
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