  Given a surjective endomorphism $f: X \to X$ on a projective variety over a
number field, one can define the arithmetic degree $\alpha_f(x)$ of $f$ at a
point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any
forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree
$\alpha_f(x)$ is strictly smaller than the first dynamical degree $\delta_f$ of
$f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic
Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small
arithmetic degree is not Zariski dense, and verify this sAND Conjecture for
endomorphisms on projective varieties including surfaces, HyperK\"ahler
varieties, abelian varieties, Mori dream spaces, simply connected smooth
varieties admitting int-amplified endomorphisms, smooth threefolds admitting
int-amplified endomorphisms, and some fibre spaces. We show the equivalence of
the sAND Conjecture and another conjecture on the periodic subvarieties of
small dynamical degree; we also show the close relations between the sAND
Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on
endomorphisms of projective spaces and another long standing conjecture on
Uniform Boundedness of torsion points in abelian varieties.
