Best known (state of science) time complexity of an array access problem

Question

Consider an Array $A$ with $n$ values and the following operations:

1. get(i): Returns the value of $A[i]$
2. insert (x): Insert the element x into the any free place in A (not necessarily in $A[x]$ or the end of the array)
3. delete (x) delete the element $A[x]$

What is the best known (amortized) time complexity of all operations? Especially the delete-op seems like it needs at least $\Omega(log n)$

See also the papers in this post. However the insert operation is defined slightly different by me, does this have any effect?