Arithmetic progressions with prescribed lp gaps
For every $n\ge3$ and $p\in[1,\infty)\setminus\{1,2,\ldots,n-1\}$, we prove that every measurable subset of $\mathbf{R}^d$ of positive upper Banach density, with $d\ge2^{n-2}+1$, contains an $n$-term arithmetic progression with every sufficiently large prescribed $\ell^p$ gap.
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