Can someone explain this notation: $\sum_{i=1}^n r_i b_i: r_i \in \mathbb{Z}(1 \leqq i \leqq n)$

Question

Can anyone explain what this means:

$\sum_{i=1}^n r_i b_i: r_i \in \mathbb{Z}(1 \leqq i \leqq n)$

I'm just having trouble understanding the second part. My understanding so far is that $r_i$ is an element of the set of integers $\mathbb{Z}$ but I don't understand the $\leqq$ sign in this context.

Answer 1

We can unfold the notation to

$$ r_1b_1 + r_2b_2 + r_3b_3 + \cdots + r_nb_n $$ where $r_1\in\mathbb Z$ and $r_2\in\mathbb Z$ and $r_3\in\mathbb Z$ and ... and $r_n\in\mathbb Z$.

In context it must be a claim that there exist particular integers $r_1, r_2, \ldots, r_n$ such that the sum in the first line satisfies whatever the context says about it, as a function of $b_1$ up to $b_n$.

(It's not a particular nice notation. Unless it's a conference submission with a strict space limit, would it have killed the authors to use a word or two of prose to clarify the relation between the variables?)

Answer 2

Here is the relevant portion of the paper:

The expression you gave appears in the context of set-builder notation. Thus, the colon should be read as "such that". It is often useful to use a vertical bar ($\LaTeX$: \mid) for better spacing.

In general, the expression $\{ x \mid \Phi(x)\}$ or $\{x : \Phi(x) \}$ is read "all elements $x$ such that $\Phi(x)$ is true".