Let S be a set of integers and denote the characteristic function of S as \chi_{S}(n). Define an operator on the space of trig functions by the relation \hat{Tf}(n) = \chi_{S}(n) \hat{f}(n). Here \hat{f}(n) is the n-th Fourier coefficient of f.

For p>=2 we'll call S a L^{p} multiplier set (or just L^{p} multiplier)  if there is an inequality of the form ||Tf||_{p} \leq c ||f||_{p}. If this inequality holds for some p but fails for p+\epsilon for every epsilon>0, we'll say that S is a strict L^{p} multiplier.

Note that every set is a L^2 multiplier and if S is a L^{p} multiplier for some p then it is a L^{q} multiplier for 2 <= q <= p. Moreover, it follows from a result of Zygmund that almost every (in the obvious sense) set is a strict L^{2} multiplier. (I also think you can prove this via Khintchine's inequality, but I haven't checked this argument.) 

Do strict L^{p} multiplier sets exist for every p>2? Note that this is similar to the \Lambda(p) problem, however, I don't see how to transform a strict \Lambda(p) set into a strict L^{p} multiplier set.