Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials.#R##N#Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT.  show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR.#R##N##R##N#Unlike related prior work, our result implies strong approximate degree lor bounds even for many functions F that have low threshold degree. Our proof is constructive:  exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function.  describe several applications, including improved separations beten the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).
['Justin Thaler']
I am writing a research paper on a idea Lower Bounds for the Approximate Degree of Block-Composed Functions.
[{"from": "human", "value": "I am writing a research paper on a idea Lower Bounds for the Approximate Degree of Block-Composed Functions."}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials.#R##N#Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT.  show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR.#R##N##R##N#Unlike related prior work, our result implies strong approximate degree lor bounds even for many functions F that have low threshold degree. Our proof is constructive:  exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function.  describe several applications, including improved separations beten the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007)."}]