Simplest exponential sum

Today‘s exponential sum curve is simply a triangle.


But yesterday‘s curve was more complex


and tomorrow‘s curve will be more complex as well.


Why is today’s curve so simple? The vertices of the curves are the partial sums of the series

\sum_{n=0}^N \exp\left( 2\pi i \left( \frac{n}{m} + \frac{n^2}{d} + \frac{n^3}{y} \right ) \right )

where m is the month, d is the day, and y is the last two digits of the year. Typically the sums are too complicated to work out explicitly by hand, but today’s sum is fairly simple. We have m = 9, d = 6, and y = 18. The three fractions add to (2n + 3n² + n³)/18. Reduced mod 18, the numerators are

0, 6, 6, 6, 12, 12, 12, 0, 0, 0, 6, 6, 6, 12, 12, 12, 0, 0

The repetition in the terms of the sum leads to the straight lines in the plot. The terms in the exponential sum only take on three values, the three cube roots of 1. These three roots are 1, a, and b where

a = exp(2πi/3) = -1/2 + i√3/2


b = exp(-2πi/3) = -1/2 – i√3/2

is the complex conjugate of a.

Using Mathematica we have

    Table[ Exp[2 Pi I (2 n + 3 n^2 + n^3)/18], {n, 0, 17}] 
        /. {Exp[2 Pi I/3] -> a, Exp[-2 Pi I/3] -> b}

which produces

1, a, a, a, b, b, b, 1, 1, 1, a, a, a, b, b, b, 1, 1

When we take the partial sums, we get four points in a straight line because they differ by a:

1, 1 + a, 1 + 2a, 1 + 3a

 then three points in a straight line because they differ by b:

(1 + 3a), (1 + 3a) + b, (1 + 3a) + 2b, (1 + 3a) + 3b

and so forth.


One thought on “Simplest exponential sum

  1. These simple exponential sums are very interesting when they have straight lines as they resemble things I have done when messing with drawing musical pitches as lines with angles. I usually set C as the vector at 0 degrees from x = 0 (trivial choice) and the 12 tone scale is simply 30 degree rotations of that vector. A series of notes is just drawing a vector with the angle of the notes pitch and then drawing the next note’s pitch off the end of the previous one. Particularly the one yesterday (5/10/20) very much resembles the shapes you create when you do this.

    I would love to turn this around and feed these sequences into a simple bit of code that converted their angle to pitch.

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