diff --git a/fragment.go b/fragment.go
index 5d89f4815..3ba4bb151 100644
--- a/fragment.go
+++ b/fragment.go
@@ -1596,7 +1596,7 @@ func (f *fragment) importPositions(set, clear []uint64, rowSet map[uint64]struct
 	var results *roaring.Bitmap
 	if smallWrite {
 		results = f.storage
-	} else if beforeCnt := f.storage.Count(); !smallWrite && beforeCnt > 0 {
+	} else if beforeCnt := f.storage.Count(); beforeCnt > 0 {
 		// Merge localBitmap into fragment's existing data.
 		if len(clear) > 0 {
 			f.storage = f.storage.Difference(clearBitmap)
@@ -1741,9 +1741,11 @@ func (f *fragment) importValue(columnIDs, values []uint64, bitDepth uint, clear
 		smallWrite = true
 		// TODO figure out how to avoid re-allocating these each time. Probably
 		// possible to store them on the fragment with a capacity based on
-		// MaxOpN.
-		toSet = make([]uint64, 0, len(columnIDs)*int(bitDepth)/2)
-		toClear = make([]uint64, 0, len(columnIDs)*int(bitDepth)/2)
+		// MaxOpN. For now, we know that the total number of bits to be
+		// set+cleared is len(values)*(bitDepth+1), so we make each slice
+		// slightly more than half of that to try to avoid reallocation.
+		toSet = make([]uint64, 0, len(columnIDs)*int(bitDepth+1)*(5/8))
+		toClear = make([]uint64, 0, len(columnIDs)*int(bitDepth+1)*(5/8))
 	}
 	f.mu.RUnlock()
 
diff --git a/roaring/roaring.go b/roaring/roaring.go
index 701f86925..5ce30b667 100644
--- a/roaring/roaring.go
+++ b/roaring/roaring.go
@@ -291,6 +291,21 @@ func (b *Bitmap) Count() (n uint64) {
 	return b.Containers.Count()
 }
 
+// Any returns "b.Count() > 0"... but faster than doing that.
+func (b *Bitmap) Any() bool {
+	iter, _ := b.Containers.Iterator(0)
+	// TODO (jaffee) I'm not sure if it's possible/legal to have an empty
+	// container, so this loop may be totally uneccesary. In theory, any empty
+	// container should be removed from the bitmap though.
+	for b := iter.Next(); b; iter.Next() {
+		_, c := iter.Value()
+		if c.n > 0 {
+			return true
+		}
+	}
+	return false
+}
+
 // Size returns the number of bytes required for the bitmap.
 func (b *Bitmap) Size() int {
 	numbytes := 0