Discrepancy
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
Points too even to be random
Independent random points clump, and the clumping is what makes the error fall only as the square root. Points chosen to be evenly spread rather than independently beat that rate, and the price is that nothing about them is random at all.
Points on a lattice that see almost nothing
Average a function over the points of a carefully tilted lattice and the error can fall like one over the square of the number of points — far faster than random sampling, and faster than the most evenly spread sequences. The reason is that a lattice rule is blind to only a thin set of frequencies, its dual lattice, and a smooth periodic function has almost nothing there.
How evenly the fractions spread
List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.
What a jump costs evenly spread points
Scrambled Sobol' points integrate a smooth function with an error that falls like N to the power −3/2. Put a jump in the integrand along a tilted line and the exponent halves, to −3/4, because about √N of the N boxes the points fill are cut by the jump and each of those is left to chance. A jump along an axis costs almost nothing, the cost grows with the length of the jump and the dimension, and integrating across the jump before sampling gives the whole rate back.
Which way a prime's two squares point
A prime one more than a multiple of four is a sum of two squares in exactly one way, and the two squares make a point on a circle. Draw that point for every such prime and ask which way it faces. It faces every way equally — the angles spread evenly over the sector, which Hecke proved by giving each angle a remainder and copying Dirichlet. Measured over seventy-four thousand primes, they are even more evenly spread than random angles would be.
Named alongside it
The objects these essays reach for when they reach for this one.
EquidistributionConvergence rateLatticeNumerical integrationQuasi-monte carloCurse of dimensionalityDirichlet theoremDual latticeError boundFarey sequenceFourier seriesGaussian integers