  We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the
3SUM conjecture) of several problems related to finding Abelian square and
additive square factors in a string. In particular, we conclude conditional
optimality of the state-of-the-art algorithms for finding such factors.
  Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of
an odd half-length, (b) computing centers of all Abelian square factors, (c)
detecting an additive square factor in a length-$n$ string of integers of
magnitude $n^{\mathcal{O}(1)}$, and (d) a problem of computing a double 3-term
arithmetic progression (i.e., finding indices $i \ne j$ such that
$(x_i+x_j)/2=x_{(i+j)/2}$) in a sequence of integers $x_1,\dots,x_n$ of
magnitude $n^{\mathcal{O}(1)}$.
  Problem (d) is essentially a convolution version of the AVERAGE problem that
was proposed in a manuscript of Erickson. We obtain a conditional lower bound
for it with the aid of techniques recently developed by Dudek et al. [STOC
2020]. Problem (d) immediately reduces to problem (c) and is a step in
reductions to problems (a) and (b). In conditional lower bounds for problems
(a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it
using several string gadgets that include arbitrarily long Abelian-square-free
strings.
  Our reductions also imply conditional lower bounds for detecting Abelian
squares in strings over a constant-sized alphabet. We also show a subquadratic
upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].
