Proving that Spec(α) and Spec(ß) partition positive integers iff α and ß are irrational and 1/α + 1/ß = 1 From Concrete Math, problem 3.13 asks:
"Let α and ß be positive real numbers. Prove that Spec(α) and Spec(ß) partition positive integers if and only if α and ß are irrational and 1/α + 1/ß = 1"
The solution claims:
"If they form a partition, the text's formula for N(α,n) implies that 1/α + 1/ß = 1, because the coefficients of n in the equation N(α,n) + N(ß,n) = n must agree if the equation is to hold for large n" (it goes on to the next part of the proof but I only care about this part)
In this chapter, they define Spec(α) to mean an infinite multiset of integers: $\{\lfloorα\rfloor,\lfloor2α\rfloor, ...\}$, and define N(α,n) to be the number of elements in Spec(α) that are $\le$ n. They show that N(α,n) = $\lceil(n+1)/α\rceil - 1$. They also show that a necessary condition for Spec(α) and Spec(ß) to partition the positive integers is N(α,n) + N(ß,n) = n  
I can write out the equation N(α,n) + N(ß,n) = n by substituting that equation:
$$\lceil(n+1)/ß\rceil + \lceil(n+1)/α\rceil - 2 = n$$
then converting to floor
$$\lfloor (n+1)/ß\rfloor + \lfloor (n+1)/α\rfloor = n$$
then splitting the floor into the fractional and actual part of its arguments
$$n(1/ß + 1/α) + 1/α + 1/ß - \{(n+1)/ß\} - \{(n+1)/α\} = n$$
...but, at that point, I don't see how I can conclude that 1/α + 1/ß = 1. I see that 1/α + 1/ß appears as a coeffecient of n, but there's the problem of the fractional stuff on the right. For their claim to be correct, the fractional parts would have to add up to 1 to cancel out the 1 from  1/α + 1/ß. I know that the fractional parts both have values less than one (since they are fractional parts of real numbers), but I don't see how to conclude that they sum to 1. Can I just claim that, since n appears on the right hand side with a coefficient of 1, that the coefficient for n on the left must be 1?

The set of positive integers equal to $[nx]$ for some $n$ (where $x > 1$ to avoid the trivial situation of covering all positive integers) has density $\frac{1}{x}$. If every number is covered exactly once by the $\alpha$ and $\beta$ sets, then the sum of their densities is $1$.
http://en.wikipedia.org/wiki/Natural_density
The rest of the proof can be found at 
http://en.wikipedia.org/wiki/Beatty_sequence
