The Erdős similarity problem.

This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.

Steinhaus [St20] has proved this is false whenever $A$ is a finite set.

This conjecture is known in many special cases (but, for example, it is open when $A=\{1,1/2,1/4,\ldots\}$, which is Problem 94 on Green's open problems list). For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem. A survey of more recent progress was written by Jung, Lai, and Mooroogen [JLM24].

Erdős (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure.

Erdős and Mauldin (unpublished) claim that this is true for trapezoids in general, but fails for parallelograms (a construction showing this fails for parallelograms was provided by Kovač) [Ko23].

Kovač and Predojević [KoPr24] have proved that this is true for cyclic quadrilaterals - that is, every set with infinite measure contains four distinct points on a circle such that the quadrilateral determined by these four points has area $1$. They also prove that there exists a set of infinite measure such that every convex polygon with congruent sides and all vertices in the set has area $<1$.

Koizumi [Ko25] has resolved this question, proving that any set with infinite measure must contain the vertices of an isosceles trapezoid, an isosceles triangle, and a right-angled triangle, all of area $1$.