Bounds for Approximate Discrete Tomography Solutions

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions f : A → {0, 1} and f : A → Z having given line sums in certain directions have been analyzed. Here A was a block in Z with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that A is an arbitrary or convex finite set in Z. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of the following type: if the line sums in k directions of a function h : A → [0, 1] are known, then there exists a function f : A → {0, 1} such that its line sums differ by at most k from the corresponding line sums of h.