# Differences of the Union of Two Arithmetic Progressions and Christoffel Paths

Differences of the Union of Two Arithmetic Progressions and Christoffel Paths

Let and be two positive integers with . Define and to be the arithmetic progressions and . The set partitions the interval into subintervals, whose lengths are plotted here. The plots are periodic, so it makes sense to wrap them around a circle with a polar plot.

m

n

l=LCM(m,n)

M

N

{0,m,2m,3m,…,l}

{0,n,2n,3n,…,l}

M⋃N

[0,l]

For example, if and , then and , so , whose successive differences are the lengths .

m=3

n=5

M={0,3,6,9,12,15}

N={0,5,10,15}

M⋃N={0,3,5,6,9,10,12,15}

{3,2,1,3,1,2,3}

If is close to a fraction with low numerator and denominator, the points of the Cartesian plot seem to lie on a net of lines, and in the special case when , the line plot looks like beats.

m/n

m/n≈1

The upper and lower Christoffel paths are paths on the integer unit lattice above and below the line from to such that no point with integer coordinates lies between them unless it is on . They envelope the unit squares that intersects. Here the paths are drawn up to the point . The paths (each of length ) break where the sorted sequence of the union of and changes from one sequence to the other.

L

(0,0)

(m,n)

L

L

(m/GCD(m,n),n/GCD(m,n))

m+n

M

N