What if $|A+A|$ is proportional to $|A|^{1 + \delta}$? Let $A$ be a finite subset of the integers and $0 < \delta \leq 1/2$. Suppose that $$|A+A| \asymp |A|^{1+\delta}.$$
What examples of such an $A$ are known? Can we conclude anything in general about the structure of $A$?
 A: If $|A+A|\le K|A|$, where $K$ does not grow with $A$ (or grows very slowly), we are in the realm of Freiman's Theorem: $A$ is densely contained in a generalized arithmetic progression. However, Freiman's Theorem (even with the best known quantitative bounds) says nothing when $K=|A|^\delta$, even if $\delta$ is small.
For small $\delta$, however, there is a fairly satisfactory answer due to Bourgain (as part of his quantitative sum-product and projection theorems, see [Bourgain, Jean. The discretized sum-product and projection theorems. J. Anal. Math. 112 (2010), 193--236.])
If $b\ge 2$ is a base and $A\subset \{0,1,\ldots,b^m-1\}$ is a set, then one can identify $A$ with a subset of the full $b$-ary tree of height $m$: the vertices at distance $j$ from the root are the intervals $[ k b^{m-j},(k+1) b^{m-j})$ which intersect $A$.
Bourgain proves the following: given $\varepsilon>0$, there are $\delta>0$, $b\ge 2$ (which can be taken as large as desired) such that the following holds if $m$ is large enough:
Suppose $A\subset \{ 0,1,b^m-1\}$ and $|A+A| \le b^{\delta m}|A|$ (which is the case if $|A+A|\approx |A|^{1+\delta}$). Then $A$ contains a subset $A'$ such that:


*

*$|A'| \ge b^{\varepsilon m}|A|$ (so $A'$ is a fairly dense subset of $A$).

*The $b$-ary tree associated to $A'$ is spherically symmetric, that is, any vertex of level $j$ has the same number $N_j$ of offspring.

*Furthermore, and this is the key point, for each $j$ either $N_j=1$ or $N_j \ge b^{1-\varepsilon}$. That is, at each level the tree has either no branching or close to full branching, uniformly over all vertices of that level.


Actually Bourgain deals with more flexible trees, but it is not hard to deduce the above formulation from his work, see Corollary 3.10 in my paper [On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, https://arxiv.org/abs/1609.07802 ].
Although the above structure theorem may appear to be rather weak at first sight, it does have powerful consequences, as shown by Bourgain and also in my paper referred above. Bourgain's initial approach in [Bourgain, J. On the Erdős-Volkmann and Katz-Tao ring conjectures. Geom. Funct. Anal. 13 (2003), no. 2, 334--365] consisted in applying Freiman's Theorem at many scales, but his more recent paper, cited above, has an elementary proof relying only on the Plünnecke inequalities. The dependence of $\delta$ on $\varepsilon$ is quantitative, and roughly exponential.
If $\delta$ is not very small, I don't think much is known. Examples are integer analogs of the ternary Cantor set: let $D\subset\{0,1,\ldots,b-1\}$ and let $A\subset [0,b^m-1]$ consist of all numbers whose base $b$ expansion has only digits from $D$. In many cases it is easy to calculate or estimate the size of $A+A$. For example, if $D=\{0,1,\ldots,c-1\}$ for $c<b/2$, then $A+A$ consists of all integers in $[0,b^m-1]$ whose base $b$ expansion has only digits from $\{0,1,\ldots,2c-2\}$, so that $|A|=c^m$ and $|A+A|=(2c-1)^m = |A|^{1+\delta}$ with $\delta\approx\log 2/\log c$. If $c$ is very large, then $A$ has the tree structure predicted by Bourgain's Theorem (without passing to a subset).
