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Transformer architectures such as Vision Transformers (ViT) have proven effective for solving visual perception tasks. However, they suffer from two major limitations; first, the quadratic complexity of self-attention limits the number of tokens that can be processed, and second, Transformers often require large amounts of training data to attain state-of-the-art performance. In this paper, we propose a new multi-head self-attention (MHSA) variant named Fibottention, which can replace MHSA in Transformer architectures. Fibottention is data-efficient and computationally more suitable for processing large numbers of tokens than the standard MHSA. It employs structured sparse attention based on dilated Fibonacci sequences, which, uniquely, differ across attention heads, resulting in inception-like diverse features across heads. The spacing of the Fibonacci sequences follows the Wythoff array, which minimizes the redundancy of token interactions aggregated across different attention heads, while still capturing sufficient complementary information through token pair interactions. These sparse attention patterns are unique among the existing sparse attention and lead to an \(O(N \log N)\) complexity, where \(N\) is the number of tokens. Leveraging only 2-6% of the elements in the self-attention heads, Fibottention embedded into popular, state-of-the-art Transformer architectures can achieve significantly improved predictive performance for domains with limited data such as image classification, video understanding, and robot learning tasks, and render reduced computational complexity. We further validated the improved diversity of feature representations resulting from different self-attention heads, and our model design against other sparse attention mechanisms.