On a problem of B. Mityagin

and f(x) sin⟨x, b⟩ ∈ L(R) for all b ∈ B, (2) imply f ∈ L(R) for any measurable function f on R (here ⟨x, b⟩ denotes the inner product of x and b). He showed (for p ≥ 1) that if (i) A = αZ and B = βZ, or (ii) A = {a} and B = {b} are singletons, then (1) and (2) imply f ∈ L(R) if and only if αβ is not an integer multiple of π in case (i) and ⟨a, b⟩ is not an integer multiple of π in case of (ii). He has also conjectured the statement in