Given that the upper bound on the number of unit distances determined by n points in a real space is $ cn^\frac{4}{3} $
Give a lower bound on the min. number of unique distances.
I am really struggling with this, any thoughts appreciated. Thanks
Attempt 1:
I know that the total distances = $n^2$
Let
$$ T = Total \space Distance = n^2$$ $$ U = Unit \space Distance \le cn^\frac{4}{3} $$ $$ K = Distinct \space Distances $$
So, $\space $ T = U + K
$$ T - K \le cn^\frac{4}{3}$$ $$ n^2 - K \le cn^\frac{4}{3}$$
Therefore
$$ K \ge n^2(1-cn^\frac{2}{3}) $$
Is this correct? The big question is does the equation T = U + K hold?