The inverse sieve conjecture in rough form

Let $X\subset [N]$ be a subset. Suppose that, for every prime $p$, the reduction $X\pmod p$ occupies at most $0.9p$ residue classes. Inverse sieve conjecture, rough form. Either $X$ has extremely small cardinality, or $X$ possesses algebraic structure. This informal formulation motivates more precise finitary and infinitary versions; the precise meaning of “extremely small” and “algebraic structure” is not specified here.