2015-6 Dense sets

Let \(A\) be an unbounded subset of the set \(\mathbb R\) of the real numbers. Let \(T\) be the set of all real numbers \(t\) such that \(\{tx-\lfloor tx\rfloor : x\in A\}\) is dense in \([0,1]\). Is \(T\) dense in \(\mathbb R\)?

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