3 each bin. F igure 2 shows the A$_{CP}$ map of D + → K 0 $_{S}$π + in bins of ( p la b π , cos θ lab π , cos θ CMS D + ( s ) ). Calculating a weighted average of t he A$_{CP}$ values over the 3-D bins, we obtain A D $^{+}$→ K 0 $_{S}$π + CP = ( − 0 . 71 ± 0 . 26)% where the uncertainty originat es f rom the finite size of the D + → K 0 $_{S}$π + (0.19%) and D + s → φπ + (0.18%) samples. The χ $^{2}$/d.o.f over the 3-D bin s is found to be 31.4/24 which corresponds to 14% probab ilit y. The s tat istical precision of the D + s → K 0 $_{S}$π + sample is too lo w to allow for a 3-D correction to A D + s → K 0 $_{S}$π + rec . For this m ode we correct for asymmetries other than A$_{CP}$ w it h an inclusive correction obtained by subtract- ing A D $^{+}$→ K 0 $_{S}$π + re c from A D $^{+}$→ K 0 $_{S}$π + CP after integrating over the ent ir e ( p lab π , cos θ lab π , cos θ CMS D + ) space. The inclusive correct io n is ( − 0 . 34 ± 0 . 18)% where the uncertainty is en- tirely d u e t o the statistical uncertainty of the D + s → φπ + sample. T he value of A D + s → K 0 $_{S}$π + CP is measured to be (+5 . 45 ± 2 . 50)%, where the uncertainty is statistical only. metry other than A h + ϵ . Eq. (2) can therefore be expressed as To correct for the asymmetries other than A$_{CP}$ , we use reconstructed samples of D + s → φπ + and D 0 → K $^{−}$π + decays and assume that A$_{CP}$ in CF decays is negligibly small at the current experimental sensitivity and that A$_{FB}$ is the same for all charmed mesons. We reconstruct φ mesons via their K $^{+}$K − decay channel for D + s → φπ $^{+}$, requiring the K $^{+}$K − invariant mass to be between 1.01 and 1.03 GeV/ c $^{2}$. The dominant source of systematic uncertainty in the A D + ( s $_{)}$→ K 0 $_{S}$π + CP measurement is the uncertainty in the A D + s → φπ + rec determination, which originates from the fol- lowing sources: the statistical uncertainty of the selected D + s → φπ + sample (0.18%); the choice of the M ( K $^{+}$K $^{−}$) interval (0.03%); and the choice of binning for the 3- D map of A D + s → φπ + rec (0.03%). Another source is the choice of fitting parameters for the invariant mass dis- The measured asymmetry for D + s → φπ + is the sum of A D + s FB and A π + ϵ . Hence one can extract the A$_{CP}$ value for the K 0 $_{S}$π + final states by subtracting the measured asymmetry for D + s → φπ + from that for D + ( s ) → K 0 $_{S}$π $^{+}$. The subtraction is performed in bins of π + momentum, p lab π , and polar angle in the laboratory system, cos θ lab π (because A h + ϵ depends on these two variables while it is uniform in azimuthal angle), and the charmed meson’s polar angle in the center-of-mass system, cos θ CMS D + ( s ) (since cos θ CMS D + ( s ) is correlated with cos θ lab π and A D + ( s ) FB depends on it). The choice of the three-dimensional (3-D) binning is selected in order to avoid large statistical fluctuations in FIG. 2: Measured A CP values for D + → K 0 $_{S}$π + in bins of ( p lab π , cos θ lab π , cos θ CMS D + ). Empty bins where no entries are plotted have no statistics. FIG. 1: Invariant mass distributions for the K 0 $_{S}$π ± and K 0 $_{S}$K ± final states. Points with error bars (note the small size of them due to the large sample) show the data and the histograms show the results of the parameterizations of the data. Signal, peaking background, and random combinatorial background components are also shown. ) 2 ) (GeV/c + S 0 M(K 1.85 1.9 1.95 2 ) 2 Events/(1 MeV/c 0 10000 20000 data signal + K S 0  K s + D random ) 2 ) (GeV/c − S 0 M(K 1.85 1.9 1.95 2 0 10000 20000 − K S 0  K s − D ) 2 ) (GeV/c + K S 0 M(K 1.85 1.9 1.95 2 ) 2 Events/(1 MeV/c 0 5000 10000 15000 data signal + S 0  K + D random ) 2 ) (GeV/c − K S 0 M(K 1.85 1.9 1.95 2 0 5000 10000 15000 − S 0  K − D −1 −0.5 0 0.5 1 + S 0  K + D CP A −0.2 −0.1 0 0.1 0.2 <0.15 lab  cos [0.5,1.0] GeV/c lab p −1 −0.5 0 0.5 1 + π$_{S}$ 0  K + D CP A −0.2 −0.1 0 0.1 0.2 <0.15 lab  cos [1.0,2.0] GeV/c lab p + D CMS cos −1 −0.5 0 0.5 1 + π$_{S}$ 0  K + D CP A −0.2 −0.1 0 0.1 0.2 <0.15 lab  cos >2.0 GeV/c lab p −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 [0.15,0.65] lab  cos [0.5,1.0] GeV/c lab p −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 [0.15,0.65] lab  cos [1.0,2.0] GeV/c lab p + D CMS cos −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 [0.15,0.65] lab  cos >2.0 GeV/c lab p −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 >0.65 lab  cos [0.5,1.0] GeV/c lab p −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 >0.65 lab  cos [1.0,2.0] GeV/c lab p + D CMS cos −1 −0.5 0 0.5 1 −0.2 −0.1 0 0.1 0.2 >0.65 lab  cos >2.0 GeV/c lab p A X $^{+}$→ K 0 $_{S}$h + rec = A X $^{+}$→ K 0 $_{S}$h + CP + A X + FB + A h + ϵ . (3)