Loaded dice with a uniformly distributed sum
Show that if a sum of two independent integer random variables is uniform on its support, then each summand is uniform on its own support.
问题
开放问题和研究问题，每个都是一个 git 仓库：很多来自精选合集，并注明出处。路线是任何人都能开启或接续的分支；检验通过后，结果才算数。
符合的问题
Rado's boundedness conjecture
Show that a linear equation that some finite colouring of the integers avoids can be avoided with a number of colours depending only on its length.
The polynomial Bogolyubov conjecture
Show that if A has density α in F_2^n, then 10A contains a coset of a subspace of codimension O(log(1/α)).
Primes p for which p − 2 has an odd number of prime factors
Prove that for infinitely many primes p, the number p − 2 has an odd number of prime factors counted with multiplicity.
Lipschitz graphs without three-term progressions
Find a function N → Z with bounded steps whose graph has no three-term progression, or prove none exists (the additive square problem).
Composite numbers in sumsets of sparse sets
Show that if A, B ⊆ {1,…,N} each have at least N^0.49 elements, then the sumset A + B contains a composite number.
Carbery's rectangle problem
Show that an open subset of the unit square with area α contains the corners of an axis-parallel rectangle of area at least cα².