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Let $a_1\ge a_2\geq\cdots\geq a_n$ be real numbers. And let $r=(r_1,\ldots,r_n)$ be sequence of random variables taking on values $1$ and $-1$ and such that $\sum_{i=1}^n r_i=0$.

I am wondering if one can estimate from above and from below $\cos\left(\sum_{i=1}^na_ir_i\right)$

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    Yes, I would like to get bound which would depend on $a_i, i=1, \ldots n$2012-05-07

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This is just an observation, but doesn't fit neatly into a comment.

$n$ must be even, otherwise it is impossible to have the sum of $r_i$ be zero.

Let $\sigma$ be a permutation so that $|a_{\sigma_1}|\geq ... \geq |a_{\sigma_n}|$. Then we have $| \sum_{i=1}^n a_i r_i | \leq |a_{\sigma_1}|+...+|a_{\sigma_{\frac{n}{2}}}|-(|a_{\sigma_{\frac{n}{2}+1}}| +...+ |a_{\sigma_n}|)$ and the bounds are achieved with appropriate (legal) choice of $r_i$. I have no idea how this translates into bounds on $\cos(\sum_{i=1}^n a_i r_i)$ without more information about the $a_i$.