What sizes would be the best to use for images: background.png, background@2x.png and background@3x.png if we want to use this image for example to cover the full width and
i think we should use different size of background images for different devices. Just use @3x scale images for background.
You can detect device with below lines.
#define IS_IPAD (UI_USER_INTERFACE_IDIOM() == UIUserInterfaceIdiomPad)
#define IS_IPHONE (UI_USER_INTERFACE_IDIOM() == UIUserInterfaceIdiomPhone)
#define IS_RETINA ([[UIScreen mainScreen] scale] >= 2.0)
#define SCREEN_WIDTH ([[UIScreen mainScreen] bounds].size.width)
#define SCREEN_HEIGHT ([[UIScreen mainScreen] bounds].size.height)
#define SCREEN_MAX_LENGTH (MAX(SCREEN_WIDTH, SCREEN_HEIGHT))
#define SCREEN_MIN_LENGTH (MIN(SCREEN_WIDTH, SCREEN_HEIGHT))
#define IS_IPHONE_4_OR_LESS (IS_IPHONE && SCREEN_MAX_LENGTH < 568.0)
#define IS_IPHONE_5 (IS_IPHONE && SCREEN_MAX_LENGTH == 568.0)
#define IS_IPHONE_6 (IS_IPHONE && SCREEN_MAX_LENGTH == 667.0)
#define IS_IPHONE_6P (IS_IPHONE && SCREEN_MAX_LENGTH == 736.0)