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!