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For every integer $n>1$, there is a prime number strictly between $n$ and $2n$. This supplies cyclic prime moduli comparable to an integer interval's length in additive combinatorics.

For every $\delta >0$, every sufficiently long integer interval has a nonconstant three-term arithmetic progression in each subset of subset density at least $\delta$. This is a theorem about additive combinatorics, distinct from the Roth theorem on approximation of algebraic irrational numbers. The Roth density-increment step proves it by repeatedly increasing subset density on a shorter arithmetic progression.