Let A $\subset [0:2]^n$, where $[0:2]=\{0,1,2\}$, then define $2A= \{ a+b\mid a,b \in A \}$. I wanted to know the best known lower-bound estimates for $|2A|$.


I intuitively expect that $|2A| \geq |A|^{\log\_3{5}}$ as this estimate works for $n=1$ and if we take $A= [0:2]^n$, then $|2A| = [0:4]^n$ and the bound fits. For $A \subset [0:d]^n$, I expect $|2A| \geq |A|^{\log\_{d+1}{(2d+1)}}$. Are there any known results for such settings?



Estimate of Minkowski sum
This is only a partial answer to your question; I believe there is more current work, and have forwarded your question to someone working in this area to see if they have more recent results.


In Theorem 5 of


*Bourgain, Jean; Dilworth, Stephen; Ford, Kevin; Konyagin, Sergei; Kutzarova, Denka*, [**Explicit constructions of RIP matrices and related problems**](https://arxiv.org/abs/1008.4535), Duke Math. J. 159, No. 1, 145-185 (2011). [ZBL1236.94027](https://zbmath.org/?q=an:1236.94027).


it is shown (in your notation) that $|2A| \geq |A|^{2\tau}$ for $A \subset [0:d]^n$, where $\tau$ solves the equation
$$ (\frac{1}{d+1})^{2\tau} + (\frac{d}{d+1})^\tau = 1.$$
They do not believe this result to be sharp, and (as you do) conjecture that $|2A| \geq |A|^{\log\_{d+1}(2d+1)}$ instead. This is known for $d=1$, see


*Woodall, D. R.*, [**A theorem on cubes**](https://doi.org/10.1112/S0025579300008913), Mathematika, London 24, 60-62 (1977). [ZBL0349.05010](https://zbmath.org/?q=an:0349.05010).


From the method of compressions one may assume without loss of generality that $A$ is a downset. If one then restricts $2A$ to the set $[0:d]^n$ then a sharp answer to your question was worked out in


*Bollobás, Béla; Leader, Imre*, [**Sums in the grid**](https://doi.org/10.1016/S0012-365X(96)00303-2), Discrete Math. 162, No. 1-3, 31-48 (1996). [ZBL0872.11007](https://zbmath.org/?q=an:0872.11007).


however I do not see an easy way to pass from this restricted sumset problem to the full sumset problem. Nevertheless, the method of compressions may be a promising technique to attack the problem.


There are related results in


*Matolcsi, Dávid; Ruzsa, Imre Z.; Shakan, George; Zhelezov, Dmitrii*, [**An analytic approach to cardinalities of sumsets**](https://doi.org/10.1007/s00493-021-4547-0), Combinatorica 42, No. 2, 203-236 (2022). [ZBL1513.11022](https://zbmath.org/?q=an:1513.11022).


which generalize a related inequality $|2A + \{0,1\}^n| \geq 2^n |A|$ that I worked out with Ben Green in


*Green, Ben; Tao, Terence*, [**Compressions, convex geometry and the Freiman-Bilu theorem**](https://doi.org/10.1093/qmath/hal009), Q. J. Math. 57, No. 4, 495-504 (2006). [ZBL1160.11003](https://zbmath.org/?q=an:1160.11003).


and the analogous problem for various additive energies was studied in [this recent paper of de Dios, Greenfeld, Ivanisvili, and Madrid](https://arxiv.org/abs/2112.09352), which would give some lower bounds on $|2A|$, but probably not the optimal ones.


