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

Question 1: For that whole $ A $ we have: $ s_3 (A) geq (d + 2) s_2 (A) – binom {d + 2} {2} s_1 (A) + binom {d + 2} {3} $?

by $ d = 1 $ $ A $ It's just a bunch of integers. Then the inequality $ s_3 (A) geq 3s_2 (A) -3s_1 (A) + 1 $ It happens if they form an arithmetic progression (in fact, equality occurs). I did not find any other example, although it must be admitted that I have not searched too much. Of course, it's easy and classic that $ s_2 (A) geq 2s_1 (A) -1 $.

A more difficult question (?) Is:

Question 2: For that whole $ A $ 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 that of question 1. For $ d = 1 $ it reads $ s_3 (A) geq 2s_2 (A) -s_1 (A) $. In fact, I do not know of any sequence that fails this.

Any indicator of relevant and similar inequalities in the literature would be highly appreciated.