Open-problem statements, with their sources attached.

What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?

Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\gt c \alpha^2$?

Let A ⊂ R be a set of positive measure. Does $A$ contain an affine copy of {1, 1/2, 1/4, . . . }?