Part V · Singer's algorithm · Lesson 15

Many sets, few shapes

Multipliers, equivalence, and the distinct Singer sets for each q.

This lesson is planned. Here is what it will cover.

You’ll learn

• For $t$ coprime to $v$, $t \cdot D$ is again a difference set.
• Equivalence under translation, negation, and multipliers.
• Hall’s multiplier theorem: multiplying by $p$ gives a translate of $D$.
• Counting the classes: $\varphi(q^2 + q + 1) / 3e$ up to translation, conjectured by Singer (1938) and proved by Berman (1953) and by Halberstam and Laxton (1964), OEIS A335866; half that, $\varphi(q^2 + q + 1) / 6e$, when mirror images also count as the same.

See every class for a given $q$ in the Singer classes gallery. The class counts $1, 2, 1, 5, 6, 4, 6, 18, 20, \dots$ for $q = 2, 3, 4, 5, 7, 8, 9, 11, 13, \dots$ appear in OEIS as A058241.