Addition theorems on the cyclic groups of order pℓpℓ

Let pp be a prime number and ℓℓ be any positive integer. Let GG be the cyclic group of order pℓpℓ and let SS be any sequence in GG of length pℓ+kpℓ+k for some positive integer k⩾pℓ-1-1k⩾pℓ-1-1 such that SS do not admit a subsequence of length pℓpℓ whose sum is zero in GG. Then we prove that there exists an element of GG which appears in SS at least k+1k+1 times.