  The popular 3-SUM conjecture states that there is no strongly subquadratic
time algorithm for checking if a given set of integers contains three distinct
elements that sum up to zero. A closely related problem is to check if a given
set of integers contains distinct $x_1, x_2, x_3$ such that $x_1+x_2=2x_3$.
This can be reduced to 3-SUM in almost-linear time, but surprisingly a reverse
reduction establishing 3-SUM hardness was not known.
  We provide such a reduction, thus resolving an open question of Erickson. In
fact, we consider a more general problem called 3-LDT parameterized by integer
parameters $\alpha_1, \alpha_2, \alpha_3$ and $t$. In this problem, we need to
check if a given set of integers contains distinct elements $x_1, x_2, x_3$
such that $\alpha_1 x_1+\alpha_2 x_2 +\alpha_3 x_3 = t$. For some combinations
of the parameters, every instance of this problem is a NO-instance or there
exists a simple almost-linear time algorithm. We call such variants trivial. We
prove that all non-trivial variants of 3-LDT are equivalent under subquadratic
reductions. Our main technical contribution is an efficient deterministic
procedure based on the famous Behrend's construction that partitions a given
set of integers into few subsets that avoid a chosen linear equation.
