Furstenberg–Sárközy theorem

The Furstenberg–Sárközy theorem is a result in additive number theory on square-difference-free sets, named after Hillel Furstenberg and András Sárközy. It states that, if ${\displaystyle S}$ is a set of natural numbers with the property that no two numbers in ${\displaystyle S}$ differ by a square number, then the natural density of ${\displaystyle S}$ is zero. That is, for every ${\displaystyle \varepsilon >0}$, and for all sufficiently large ${\displaystyle n}$, the fraction of the numbers up to ${\displaystyle n}$ that are in ${\displaystyle S}$ is less than ${\displaystyle \varepsilon }$. Equivalently, every set of natural numbers with positive upper density contains two numbers whose difference is a square.

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