Praise be to Allah.
It seems that the questioner is referring to the kind of chewing gum which is well known nowadays, which is a rubbery substance that usually contains sugar and artificial flavourings. Chewing this kind of gum leads to breaking the fast, because the sugars and flavourings dissolve in the saliva and enter the stomach. Undoubtedly this breaks one’s fast, because nutrients enter the stomach. But if the gum does not contain any substance that will dissolve and enter the stomach, then it does not break the fast.
With regard to alternatives to chewing gum, if what is meant is something that may be used to improve the smell of the mouth, then the Muslim should use siwaak, because it is one of the Sunnahs narrated from the Prophet (peace and blessings of Allaah be upon him). He may also rinse his mouth so as to remove the smell, or use toothpaste, subject to the condition that nothing of it reaches his stomach. If he fears that some of it may reach his stomach then he should not use it.
You should also note that this smell that comes out of the mouth because the stomach is empty and which may disturb a person, cannot be removed with the siwaak etc because it comes from the stomach due to fasting. It is better before Allaah than the fragrance of musk. It was narrated from Abu Hurayrah that the Prophet (peace and blessings of Allaah be upon him) said: “The smell of the fasting person’s mouth is better before Allaah than the fragrance of musk.” Narrated by al-Bukhaari, 5583; Muslim, 1151. See also question no. 22913.
But if the purpose of using gum is to treat the jaw by means of moving it, please see question no. 38552.
And Allaah knows best.