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Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ball $B(0,1)=\{x\in\mathbb R^n:\;|x|\leq1\}$.

I could prove that every functions $b_d$ are in $BMO(\mathbb R^n)$ space.

For the definition of BMO space, please:
or in a recent post of mine on mathstackexchange: Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

In a paper, authors said that: by the Riemann-Lebesgue Lemma, we can take $d$ very large, depending on $\psi$, so that (there exists $d$ so that) $ b(x)\int_0^1\psi(t)dt-\int_0^{\min\{1,1/|x|\}}\psi(t){\rm sgn}(\sin \pi d\cdot t\cdot|x|)dt\geq\frac12\int_0^1\psi(t)dt$ for all $\frac 12<|x|<1$. (1)

But as I know, the Rieamann-Lebesgue Lemma:

then it should applying for $\sin ...$ function not for ${\rm sgn}(t)$ function (see here for its definition: ).

So my questions are:

How should I understand these above claims (about Rieamnn-Lebesgue lemma) in the paper? Do functions $b_d$ should be $b_d(x)=1_{B(0,1)}\cdot\sin (\pi d|x|)$. If that, are $b_d$ in BMO? If the new functions $b_d$ in BMO, does (1) hold for some large $d$?

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