nt.number theory – In the set \$ { sum_ {k = 1} ^ n lambda_ka_k: a_1, ldots, a_k text {are elements other than} A } \$

For a field $$F$$ leave $$p (F) = p$$ If the characteristic of $$F$$ Is a cousin $$p$$Y $$p (F) = + infty$$ Yes $$F$$ It is zero characteristic.

In 2007, I considered the linear extension of the Erdos-Heilbronn conjecture, and I surmised that (see http://arXiv.org/abs/0810.0467) for any non-zero element $$lambda_1, ldots, lambda_n$$ of a field $$F$$ with $$p (F) not = n + 1$$ and a finite subset $$A$$ of $$F$$ we have
begin {align} & | { lambda_1a_1 + ldots + lambda_n a_n: a_1, ldots, a_n text {are elements other than} A } | \ & qquad qquad ge min {p (F) – delta, , n (| A | -n) +1 }, end {align}
where $$delta$$ is $$1$$ Yes $$n = 2$$ Y $$lambda_1 + lambda_2 = 0$$Y $$delta = 0$$ otherwise.

Motivated by the above and by question 316142 mine, here I pose the following question.

QUESTION: Is my next true conjecture?

Guess. Leave $$lambda_1, ldots, lambda_n (n ge3)$$ be positive integers with $$lambda_1 le ldots le lambda_n le n$$ Y $$gcd ( lambda_1, ldots, lambda_n) = 1$$. Leave $$F$$ be a field with $$p (F)> n + 1$$. So, for any finite subset $$A$$ of $$F$$ with $$| A | ge n + delta_ {n, 3}$$ we have
begin {align} & bigg | bigg { sum_ {k = 1} ^ n lambda_ka_k: a_1, ldots, a_n text {are elements other than} A bigg bigg | ge & min bigg {p (F), ( lambda_1 + ldots + lambda_n) (| A | -n) + sum_ {k = 1} ^ { lfloor n / 2 rfloor} (n + 1-2k) ( lambda_ {n + 1-k} – lambda_k) +1 bigg }. end {align}

Now let me explain where the lower limit comes from. Suppose that $$A$$ it's just the subset $${1, ldots, m }$$ of the rational field $$mathbb Q$$. For the whole
$$S = { lambda_1a_1 + ldots + lambda_na_n: a_1, ldots, a_n text {are elements other than} A },$$
its minimum element must be $$sum_ {k = 1} ^ n lambda_k (n + 1-k)$$ While its maximum element should be $$sum_ {k = 1} ^ n lambda_k (m-n + k)$$. Note that
begin {align} & bigg | bigg { sum_ {k = 1} ^ n lambda_k (n + 1-k), ldots, sum_ {k = 1} ^ n lambda_k (m-n + k) bigg } bigg | \ = & ( lambda_1 + ldots + lambda_n) (mn) + sum_ {k = 1} ^ { lfloor n / 2 rfloor} (n + 1-2k) ( lambda_ {n + 1- k) – lambda_k) +1. end {align}
Yes $$lambda_k = k$$ for all $$k = 1, ldots, n$$, so
$$sum_ {k = 1} ^ { lfloor n / 2 rfloor} (n + 1-2k) ( lambda_ {n + 1-k} – lambda_k) = sum_ {k = 1} ^ { lfloor n / 2 rfloor} (n + 1-2k) ^ 2 = frac {n (n ^ 2-1)} 6.$$

Any comment is welcomed!