math.CA — Classical Analysis and ODEs

Arithmetic progressions with prescribed lp gaps

Contributed by Vjekoslav Kovač

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.