Logarithmic selection of an iteration level

Statement

Let $T>1$ , $A\ge1$ , and define $i=\lfloor\log_T A\rfloor$ , where $\log_T A=(\log A)/(\log T)$ . Then

A/T<T^i\le A.

The resulting quantities are comparable with constants depending on the fixed base. This follows directly from $i\le\log_T A<i+1$ . For $\Lambda>1$ and $\rho=\log\Lambda/\log T$ , it follows that

\Lambda^{-1}A^\rho<\Lambda^i\le A^\rho.

The integer iterate therefore matches the desired size up to fixed multiplicative constants.

Neighboring levels

For two positive targets $A,B\ge1$ , their selected levels obey

|i(A)-i(B)|\le1+\frac{|\log(A/B)|}{\log T}.

Thus a uniform bound on target ratios gives a uniform bound on level differences. The level is discrete and locally constant except at threshold values; no derivative of the floor function is used in smooth coordinate estimates.