N}}}\mapsto (b_{\log_2(n)}\chi_{2^{{\mathbb N}}}(n))_{n\in {{\mathbb N}}}$. Here $2^{{\mathbb N}}=\{1,2,4,8,16,\ldots\}$ denotes the dyadic natural numbers. It should be pointed out that the ideal $\mathfrak{m}_{\tilde{\psi}}({{\mathbb N}})$ supports Dixmier traces defined directly from $\tilde{\psi}$ if and only if $A_\psi(e)=1$ (in which case $\tilde{\psi}$ has regular variation of index $k_\psi=0$).

Summing up, there are constants $\alpha_0,\alpha_1>0$ such that for any exponentiation invariant extended limit $\omega$, and $c\in \mathfrak{m}_{\tilde{\psi}}^{(p)}$ $$\label{estofrtrTr}
\alpha_0 \mathrm{tr}_{\omega\circ \log_2,\tilde{\psi}}(|c|^p)\leq \mathrm{Tr}_{\omega,\psi}(|H_{\overline{\gamma_{p,c}}}|^p)\leq \alpha_1 \mathrm{tr}_{\omega\circ \log_2,\tilde{\psi}}(|c|^p).$$

For a function $g\in L^\infty(0,\infty)$ define the function $\bar{g}\in L^\infty(0,\infty)$ by the formula $$\bar{g}(t):=\int_{\lfloor t \rfloor }^{\lfloor t \rfloor+1}g(s)\mathrm{d} s.$$ Set $C:=-\liminf_{t\to \infty} g(t)$ and define $$\label{cnfromg}
c_n:=\left( |\bar{g}(n)+C|\cdot \tilde \psi'(n)\right)^{1/p}, \ n\ge0.$$ It is easy to see that $c=|c|\in \mathfrak{m}_{\tilde{\psi}}^{(p)}({{\mathbb N}})$.

\[integracom\] Assume that $g\in L^\infty(0,\infty)$ for some $\beta>0$ satisfies that $$\label{conditionongorfor}
g(t)-\bar{g}( t)= O(t^{-\beta}), \quad \mbox{as $t\to \infty$}.$$ For $c=(c_n)_{n\in {{\mathbb N}}}\in\mathfrak{m}^{(p)}_{\tilde{\psi}}({{\mathbb N}})$ defined as in Equation , it holds that $$\mathrm{tr}_{\omega\circ \log_2,\tilde{\psi}}(|c|^p)=\lim_{t\to\omega\circ \log_2} \frac{1}{\tilde{\psi}(t)}\int_0^t g(s)\cdot \tilde \psi'(s) \mathrm{d}s+C,$$ where $C=-\liminf_{t\to \infty} g(t)$.

By the definition of liminf, we have that $g+C-|g+C|=o(1)$. It follows that the function $$\frac{1}{\tilde{\psi}(t)}\int_0^t \sum_{j=0}^\infty|\bar{g}(j)+C|\cdot \tilde \psi'(j) \chi_{(j,j+1]}(s) \mathrm{d}s - \frac{1}{\tilde{\psi}(t)}\int_0^t \sum_{j=0}^\infty(\bar{g}(j)+C)\cdot \tilde \psi'(j) \chi_{(j,j+1]}(s)\mathrm{d}s$$ is $o(1)$ as $\to\infty$. We therefore have $$\begin{aligned}
\mathrm{tr}_{\omega\circ \log_2,\tilde{\psi}}(|c|^p)&=\lim_{t\to\omega\circ \log_2} \frac{1}{\tilde{\psi}(t)}\int_0^t \sum_{j=0}^\infty|c_j|^p \chi_{(j,j+1]}(s) \mathrm{d}s\\
&=\lim_{t\to\omega\circ \log_2} \frac{1}{\tilde{\psi}(t)}\int_0^t 