Study of the structure of a tannakian category formed by equivalence classes of systems of linear inequalities over a number field

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It is known that the equivalence classes of systems of linear inequalities which describe a rational approximation property of numbers are naturally identified with the representations of some group, but the whole body of the huge group is not clear yet.In this study, we have revealed via what algebraic group a classical system of linear inequalities which corresponds to the famous Roth inequality is considered a representation of the said huge group. We have also shown that arbitrary reductive groups appear in a natural manner as quotients of the above-mentioned huge group.