Notation. The main objects of this paper are n-tuples y = (y1, . . . , yn) of real numbers viewed as linear forms, i.e. as row vectors. In what follows, y will always mean a row vector, and we will be interested in values of a linear form given by y at integer points q = (q1, . . . , qn)T , the latter being a column vector. Thus yq will stand for y1q1 + · · ·+ ynqn. Hopefully it will cause no confusion. We will study differentiable maps f = (f1, . . . , fn) from open subsets U of R to R; again, f will be interpreted as a row vector, so that f(x)q stands for q1f1(x) + · · · + qnfn(x). In contrast, the elements of the “parameter set” U will be denoted by x = (x1, . . . , xd) without boldfacing, since the linear structure of the parameter space is not significant. For f as above we will denote by ∂if : U 7→ R, i = 1, . . . , d, its partial derivative (also a row vector) with respect to xi. If F is a scalar function on U , we will denote by∇F the column vector consisting of partial derivatives of F . With some abuse of notation, the same way we will treat vector functions f : namely, ∇f will stand for the matrix function U 7→ Md×n(R) with rows given by partial derivatives ∂if . We will also need higher order differentiation: for a multiindex β = (i1, . . . , id), ij ∈ Z+, we let |β| = ∑d j=1 ij and ∂β = ∂ i1 1 ◦ · · · ◦ ∂ id d . Unless otherwise indicated, the norm ‖x‖ of a vector x ∈ R (either row or column vector) will stand for ‖x‖ = max1≤i≤k |xi|. In some cases however we will work with the Euclidean norm ‖x‖ = ‖x‖e = √∑k i=1 x 2 i , keeping the same notation. This distinction will be clearly emphasized to avoid confusion. We will denote by R1 the set of unit vectors in R (with respect to the Euclidean norm). We will use the notation |〈x〉| for the distance between x ∈ R and the closest integer, |〈x〉| def = mink∈Z |x− k|. (It is quite customary to use ‖x‖ instead, but we are not going to do this in order to save the latter notation for norms in vector spaces.) If B ⊂ R, we let |B| stand for the Lebesgue measure of B.