Find the minimum value of $m$ such that any $m$-element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le  3$.
1. **Claim**: The minimum value of \( m \) such that any \( m \)-element subset of the set of integers \(\{1, 2, \ldots, 2016\}\) contains at least two distinct numbers \( a \) and \( b \) which satisfy \( |a - b| \leq 3 \) is \( \boxed{505} \).

2. **Verification that \( 504 \) is not enough**:
    - Consider the subset \(\{4, 8, 12, \ldots, 2016\}\). This subset contains \( 504 \) elements.
    - For any two elements \( a \) and \( b \) in this subset, \( |a - b| \geq 4 \). Therefore, it is possible to have a 504-element subset where no two elements have a difference of 3 or less.

3. **Verification that \( 505 \) is enough**:
    - Partition the set \(\{1, 2, 3, \ldots, 2016\}\) into 504 subsets, each containing 4 consecutive elements:
      \[
      \{1, 2, 3, 4\}, \{5, 6, 7, 8\}, \{9, 10, 11, 12\}, \ldots, \{2013, 2014, 2015, 2016\}
      \]
    - By the Pigeonhole Principle, if we choose any 505 elements from the set \(\{1, 2, 3, \ldots, 2016\}\), at least one of these 504 subsets must contain at least 2 elements.
    - In each subset, the maximum difference between any two elements is 3. Therefore, there must exist at least two elements \( a \) and \( b \) such that \( |a - b| \leq 3 \).

Conclusion:
\[
\boxed{505}
\]