Sharpness results concerning finite differences in Fourier analysis on the circle group

Let G denote the group or , let denote the identity element of G, and let s be given. Then, a \emph{difference of order} s is a function fL2(G) for which there are aG and gL2(G) such that f=(−a)sg. Let s(L2(G)) be the vector space of functions that are finite sums of differences of order s. It is known that if fL2(), fs(L2()) if and only if −f(x)2x−2sdx . Also, if fL2(), fs(L2()) if and only if f(0)=0. Consequently, s(L2(G)) is a Hilbert space in a (possibly) weighted L2-norm. It is known that every function in s(L2(G)) is a sum of 2s+1 differences of order s. However, there are functions in s(L2()) that are not a sum of 2s differences of order s, and we call the latter type of fact a \emph{sharpness result}. In 1(L2()), it is known that there are functions that are not a sum of two differences of order one. A main aim here is to obtain new sharpness results in the spaces s(L2()) that complement the results known for , but also to present new results in s(L2()) that do not correspond to known results in s(L2()). Some results are obtained using connections with Diophantine approximation. The techniques also use combinatorial estimates for potentials arising from points in the unit cube in Euclidean space, and make use of subtraction sets in arithmetic combinatorics.