  We consider a set of transmitters broadcasting simultaneously on the same
frequency under the SINR model. Transmission power may vary from one
transmitter to another, and a transmitter's signal strength at a given point is
modeled by the transmitter's power divided by some constant power $\alpha$ of
the distance it traveled. Roughly, a receiver at a given location can hear a
specific transmitter only if the transmitter's signal is stronger by a
specified ratio than the signals of all other transmitters combined. An SINR
query is to determine whether a receiver at a given location can hear any
transmitter, and if yes, which one.
  An approximate answer to an SINR query is such that one gets a definite YES
or definite NO, when the ratio between the strongest signal and all other
signals combined is well above or well below the reception threshold, while the
answer in the intermediate range is allowed to be either YES or NO.
  We describe compact data structures that support approximate SINR queries in
the plane in a dynamic context, i.e., where transmitters may be inserted and
deleted over time. We distinguish between two main variants --- uniform power
and non-uniform power. In both variants the preprocessing time is $O(n
\mathop{\textrm{polylog}} n)$ and the amortized update time is
$O(\mathop{\textrm{polylog}} n)$, while the query time is
$O(\mathop{\textrm{polylog}} n)$ for uniform power, and randomized time
$O(\sqrt{n} \mathop{\textrm{polylog}} n)$ with high probability for non-uniform
power.
  Finally, we observe that in the static context the latter data structure can
be implemented differently, so that the query time is also
$O(\mathop{\textrm{polylog}} n)$, thus significantly improving all previous
results for this problem.
