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In this article, we address Amit--Ashurst conjecture on lower bounds of aprobability distribution associated to a word on a finite nilpotent group. Weobtain an extension of a result of Camina, Iniguez, and Thillaisundaram byproviding improved bounds for the case of finite nilpotent groups of class $2$for words in an arbitrary number of variables, and also settle the conjecturein some cases. We achieve this by obtaining words that are identicallydistributed on a group to a given word. In doing so, we also obtain animprovement of a result of Iniguez and Sangroniz using elementary techniques.

Automorphic word maps and Amit--Ashurst conjecture