N=3 supersymmetric Born-Infeld theory

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We construct an off-shell N=3 supersymmetric extension of the abelian D=4 Born-Infeld action starting from the action of supersymmetric Maxwell theory in N=3 harmonic superspace. A crucial new feature of the N=3 super BI action is that its interaction part contains only terms of the order 4k in the N=3 superfield strengths. The correct component bosonic BI action arises as the result of elimination of auxiliary tensor field which is present in the off-shell N=3 vector multiplet in parallel with the gauge field strength. In this new Legendre-type representation, the bosonic BI action is fully specified by a real function of the single variable quartic in the auxiliary tensor field. The generic choice of this function amounts to a wide set of self-dual nonlinear extensions of the Maxwell action. All of them admit an off-shell N=3 supersymmetrization.