Abstract
Let q be a prime power, and Fq be the finite field of order q. In this short note, by using methods from spectral graph theory, we give sufficient conditions to guarantee that Fq \ {0} {a1u1x1 + ... + adudxd : 1 ≤ xi ≤ Mi; 1 ≤ i ≤ d} ; where ai and ui are non-zero elements in Fq, and Mi are integers. This result generalizes a recent result given by Cilleruelo and Zumalacárregui (2014). Using the same techniques, we extend this result in the setting of the finite cyclic ring.
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