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$B_h[g]$
-vectors and finite families of linear forms
- Part of
-
- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 13 July 2026, pp. 1-6
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- Article
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In a recent work, Nathanson proved that, for $K=\mathbb {R}$
or $\mathbb {C}$
, the set of $B_h$
-vectors in $K^n$
is open and dense. Leonetti later extended this to $B_h[g]$
-vectors. In this note, we isolate a finite linear functional principle behind these results. Namely, if $\mathcal F$
is a finite family of pairwise distinct linear functionals on $K^n$
, then the set of points at which no value is assumed by more than g members of $\mathcal F$
is open and dense in $K^n$
. As consequences, we recover the theorems of Nathanson and Leonetti for $B_h[g]$
-vectors and obtain the following variant: for every linear form $L(x_1,\ldots ,x_h)=c_1x_1+\cdots +c_hx_h$
, with nonzero integer coefficients, the corresponding set of $L[g]$
-vectors, evaluated on distinct variables, is open and dense in $K^n$
.
