Bounding the Sum of Square Roots via Lattice Reduction

Understand

Let $k$ and $n$ be positive integers.

• Define $R(n,k)$ to be the minimum positive value of $$ | e_i \sqrt{s_1} + e_2 \sqrt{s_2} + ...
• + e_k \sqrt{s_k} -t | $$ where $ s_1, s_2, ..., s_k$ are positive integers no larger than $n$, $t$ is an integer and $e_i\in \{1,0, -1\}$ for all $1\leq i\leq k$.
• It is important in computational geometry to determine a good lower and upper bound of $ R(n,k)$.