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A set a1 …, ak of different residues mod v is called a difference set (v, k, λ) (v>k > λ) if the congruence ai — aj ≡ d (mod v) has exactly λ solutions for d ≢ 0 (mod v). Singer  has demonstrated the existence of a difference set (v, k, 1) if k — 1 is a prime power, and difference sets for λ > 1 have been constructed by various authors; but necessary and sufficient conditions for the existence of a (v, k, λ) are not known. It has not been possible so far to find a difference set with λ = 1 if k — 1 is not a prime power and it has therefore been conjectured that no such difference set exists.
In a recent paper I. J. Schoenberg  considered relations
where the av are rational integers and the ζv are roots of unity. We may in (1) replace all negative coefficients av by −av replacing at the same time ζv by −ζv so that we may, if it is convenient, assume that all av are positive. If we do this and arrange the ζr so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all av are non-negative. We shall call a polygon (1) k-sided if all av are positive. The polygon is called degenerate if two of the ζv are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p-gons where p is a prime.
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