Functions over finite fields that determine few directions

We investigate functions f over a finite field Fq, with q prime, with the property that the map x goes to f(x) + cx is a permutation for at least 2 √ q − 1 elements c of the field. We also consider the case in which q is not prime and f is a function in many variables and pairs of functions (f, g) with the property that the map x goes to f(x) + cg(x) + dx is a permutation for many pairs (c, d) of elements of the field. Let Fq denote the finite field with q elements and suppose that f is a function from Fq to Fq for which there are M(f) elements c ∈ Fq with the property that x 7→ f(x) + cx is a permutation of Fq. Equivalently, for all distinct x and y in Fq we have f(x)+cx 6= f(y)+cy and so −c 6= (f(y)− f(x))/(y− x), and so −c is not a direction determined by the graph of the function f . In [10] Rédei proved that if q is prime and M(f) > (q − 1)/2 then f is linear, and so M(f) = q − 1. As a corollary to this result he proved that if G is an elementary abelian group of size q and A and B are subsets of G with the property that any element of G can be written uniquely as the sum of an element of A and an element of B, then either A or B is a coset. In the articles [5] (not characteristic 2 or 3) and [1] (all characteristics) it was proved for any q that if M(f) > (q−1)/2 then either f is linear or q− q/s ≥ M(f) ≥ q+1− (q−1)/(s−1) for some subfield Fs of Fq and if s > 2 then f is linear over Fs. As mentioned above M(f) is the number of directions (not including the infinite direction) not determined by the graph of f , {(x, f(x)) | x ∈ Fq}. Clearly applying affine transformations to the graph of f does not alter M(f) and so we are only interested in functions f up to affine transformations. If q is odd and we take f(x) = x then M(f) = (q − 1)/2. Moreover, Lovász and Schrijver [8] proved that if M(f) = (q − 1)/2 and q is prime then the function f is affinely equivalent to x. This may extend to all odd q, no proof or counterexample is known. The author acknowledges the support of the Ramon y Cajal programme and the project MTM200508990-C02-01 of the Spanish Ministry of Science and Education and the project 2005SGR00256 of the Catalan Research Council.