We revisit the Gaussian broadcast channel (GBC) and explore the rate region achieved by purely discrete inputs with treating interference as noise (TIN) decoding. Specifically, we introduce a simple scheme based on superposition coding with identically and independently distributed (i.i.d.) inputs drawn from discrete constellations, e.g., pulse amplitude modulations (PAM). Most importantly, we prove that the resulting achievable rate region under TIN decoding is within a constant gap to the capacity region of the GBC, where the gap is independent of all channel parameters. In addition, we show via simulation that the weak user can achieve a higher rate with PAM than with Gaussian signaling in some cases.