VTK/C++ - coloring points to scalar values on custom PolyData to custom colors

Question

I want to to plot angular values on a sphere with evenly spaced points in C++/VTK. Therefore I want to color points on a custom VTKPolyData with selected colors. As a starting point I used https://kitware.github.io/vtk-examples/site/Cxx/Visualization/ScalarBarActor/ which creates a similiar picture.

I successfully created the custom PolyData as I wanted and calculated the values for the Points (all are positive doubles, mostly smaller than 1.0), but the coloring doesn't fit with the values I set:

    //create evenly spaced out angular coordinates Phi and Theta ...
    ...
    
    //make a sphere object
    sphere = vtkPolyData::New();
    vtkPoints* spherepoints = vtkPoints::New();
    
    //create Points on Phi and Theta to (x,y,z)
    ...spherepoints->InsertNextPoint(P);...
    
    //Add points to PolyData
    sphere->SetPoints(spherepoints);
    sphere->Allocate();
    
    //Create Connectivity in PolyData by 
    ...
    vtkIdList* indices = vtkIdList::New();
    indices->Allocate(3);
    indices->InsertId(0, integer0);
    indices->InsertId(1, integer1);
    indices->InsertId(2, integer2);
    sphere->InsertNextCell(VTK_TRIANGLE, indices);
    ...

    
    vtkFloatArray* scalars = vtkFloatArray::New();

    //for all points calculate the local value
    for (int i = 0; ...; i++) {
        double I = f(i);
        scalars->InsertTuple1(i, I);
    }
    // add scalars
    sphere->GetPointData()->SetScalars(scalars);

    //set a custom threshold
    double threshold = 1e-4;

    // build a lookuptable (low values: blue, high: red)
    vtkLookupTable* lut = vtkLookupTable::New();
    lut->SetIndexedLookup(false);
    lut->Build();
    lut->SetTableValue(0.01 * threshold, 0.0, 0.0, 1.0, 0.1);
    lut->SetTableValue(0.1 * threshold, 1.0, 1.0, 1.0, 0.3);
    lut->SetTableValue(1 * threshold, 1.0, 1.0, 0.0, 0.5);
    lut->SetTableValue(10 * threshold,1.0, 0.0, 0.0, 0.7);
    lut->SetTableRange(0, 10* threshold);

    //build mapper
    vtkNew<vtkPolyDataMapper> mapper;
    mapper->SetInputData(sphere);
    mapper->ScalarVisibilityOn();
    mapper->SetLookupTable(lut);
    mapper->SetScalarModeToUsePointData();
    mapper->SetColorModeToMapScalars();
    //create actor
    this->sphereactor = vtkSmartPointer<vtkActor>::New();
    sphereactor->SetMapper(mapper);
    sphereactor->VisibilityOn();

I also tried using only integer values (e.g. by mapping to 0-1000) for scalars, only indexing my colors (0-3), using more custom interpolating color points or using the SetTableRange or SetValueRange command with no success. The picture I get everytime is mostly one or two colors, which do or don't correspont to the scalars in a way, I can understand. What is the correct way to color the surface by the scalar values?