# co.combinatorics – Inequalities about tripling and duplicating a set

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.