A density version of Hua’s theorem

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℙ</m:mi> </m:math> {\mathbb{P}} be the set of primes. Given five sets <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>P</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> {P_{i}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>4</m:mn> <m:mo>,</m:mo> <m:mn>5</m:mn> </m:mrow> </m:mrow> </m:math> {i=1,2,3,4,5} , in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℙ</m:mi> </m:math> {\mathbb{P}} with relative lower density <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:munder accentunder="true"> <m:mi>d</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mi>ℙ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mi>ε</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:math> {\underline{d}_{\mathbb{P}}(P_{i})=\varepsilon_{i}} , we prove that given the condition that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>5</m:mn> </m:msubsup> <m:msubsup> <m:mi>ε</m:mi> <m:mi>i</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> <m:mo>&gt;</m:mo> <m:mn>4</m:mn> </m:mrow> </m:math> {\sum_{i=1}^{5}\varepsilon_{i}^{2}&gt;4} , every sufficiently large number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≡</m:mo> <m:mrow> <m:mpadded width="+2.2pt"> <m:mn>5</m:mn> </m:mpadded> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mpadded width="+3.3pt"> <m:mi>mod</m:mi> </m:mpadded> <m:mo>⁢</m:mo> <m:mn>24</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {n\equiv 5\mskip 4.0mu (\mathrm{mod}\mskip 6.0mu 24)} can be written as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mn>5</m:mn> </m:msubsup> <m:msubsup> <m:mi>p</m:mi> <m:mi>i</m:mi> <m:mn>2</m:mn> </m:msubsup> </m:mrow> </m:mrow> </m:math> {n=\sum_{i=1}^{5}p_{i}^{2}} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>i<