# Inequalities about tripling and doubling sumsets

Let $$A$$ be a set of vectors in $$\mathbb Z^d$$ who $$\mathbb R$$-span is the whole $$\mathbb R^d$$. Let $$s_i(A)$$ denote the size of $$A+A+\dots A$$ ($$i$$ times). I am interested in the following:

Question 1: Find sufficient conditions on $$A$$ so we have: $$s_3(A)\geq (d+2)s_2(A) -\binom{d+2}{2}s_1(A)+\binom{d+2}{3}$$?

For $$d=1$$ $$A$$ is just a bunch of integers. Then the inequality $$s_3(A)\geq 3s_2(A)-3s_1(A)+1$$ occurs if they form an arithmetic progression (in fact, equality happens). I did not find other examples, although admittedly I have not looked too hard (Seva pointed out in the comment that this hold for "generic" A). It is of course easy and classical that $$s_2(A)\geq 2s_1(A)-1$$.

A harder(?) question is:

Question 2: Find sufficient conditions on $$A$$ so we have: $$s_3(A)\geq (d+1)s_2(A) -\binom{d+1}{2}s_1(A)+\binom{d+1}{3}$$?

This inequality is weaker than the one in Question 1. For $$d=1$$ it reads $$s_3(A)\geq 2s_2(A)-s_1(A)$$. In fact, I do not know any sequence that fails this.

Any pointer to relevant, similar inequalities in the literature would be greatly appreciated.

• Is there any difference between your $s_i(A)$ and the quantity commonly denoted by $|iA|$ (the size of the $i$-fold sumset $iA$)? – Seva Dec 8 '18 at 14:10
• No, I am just not sure what is the standard notation. – Hailong Dao Dec 8 '18 at 14:11
• I do not understand then. For a set $A$ in a general position, one has $|3A|\sim |A|^3/6$, while $|2A|=O(|A|^2)$. Therefore, your inequalities hold true for any "typical" set, not necessarily an arithmetic progression or anything else of this sort? – Seva Dec 8 '18 at 14:33
• No, it does not. Take $A$ to be a subset of $[0,l]$ such that $2A=[0,2l]$. Then $|3A|+|A|\le 3l+|A|+1<4l+2=2|2A|$ provided $|A|<l+1$. – Seva Dec 8 '18 at 15:59
• There are lots of papers dealing with sumset inequalities, many of them due to Imre Ruzsa, but your particular inequalities hold almost always. – Seva Dec 8 '18 at 16:20