| Summary: | In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>p</mi></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>N</mi></mrow></semantics></math></inline-formula> is dependent on the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mrow><mo>(</mo><mrow><msub><mi>p</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Different sequences <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><msub><mi>p</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, for the same set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>A</mi></semantics></math></inline-formula>, will yield new and distinct densities. If the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><msub><mi>p</mi><mi>n</mi></msub></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> does not differ from the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in its order of magnitude, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfrac><mrow><msub><mi>p</mi><mi>n</mi></msub></mrow><mi>n</mi></mfrac><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then the resulting quasi-density is very close to the asymptotic density. The results for sequences that do not satisfy this condition are more interesting. In the next part, we deal with the necessary and sufficient conditions so that the quasi-statistical convergence will be equivalent to the matrix summability method for a special class of triangular matrices with real coefficients.
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