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Rudin's Conjecture
Rudin's conjecture is a mathematical hypothesis (in additive combinatorics and elementary number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...) concerning an upper bound for the number of squares in finite arithmetic progressions. The conjecture, which has applications in the theory of trigonometric series, was first stated by Walter Rudin in his 1960 paper ''Trigonometric series with gaps''. For positive integers N, q, a define the expression Q(N; q, a) to be the number of square number, perfect squares in the arithmetic progression qn + a, for n = 0, 1, \ldots, N-1, and define Q(N) to be the maximum of the set . The conjecture asserts (in big O notation) that Q(N) = O(\sqrt ) and in its stronger form that, if N > 6, Q(N) = Q(N; 24, 1). References {{reflist Combinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Combinatorics
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset ''A'' + ''B'' is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for , A + B, in terms of , A, and , B, . This can be viewed as an inverse problem with the given information that , A+B, is sufficiently small and the structural conclusion is then of the form that either A or B is the empty set; however, in literature, such problems are sometimes considered to be direct problems as well. Examples of this type include the Erdős–Heilbronn Conjecture (for a restricted sumset) and the Cauchy–Davenport Theorem. The methods used for tackling such questions often come from many different fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |