| Summary: | <p style="text-align:justify;"> Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> be a cubic form with integer coefficients in n variables, and let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> be the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>-invariant of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>. Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><mo class="MathClass-op">…</mo><mo class="MathClass-punc">,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msub></math> be linear forms with real coefficients such that, if <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi> <mo class="MathClass-rel">∈</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>r</mi></mrow></msup> <mo class="MathClass-bin">∖</mo><mrow><mo class="MathClass-open">{</mo><mrow><mn>0</mn></mrow><mo class="MathClass-close">}</mo></mrow></math>, then α⋅L is not a rational form. Assume that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi> <mo class="MathClass-rel">></mo> <mn>1</mn><mn>6</mn> <mo class="MathClass-bin">+</mo> <mn>8</mn><mi>r</mi></math>. Let <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi> <mo class="MathClass-rel">∈</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mi>r</mi></mrow></msup></math>, and let η be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi> <mo class="MathClass-rel">∈</mo> <msup><mrow><mrow><mo class="MathClass-open">[</mo><mrow><mo class="MathClass-bin">−</mo><mi>P</mi><mo class="MathClass-punc">,</mo><mi>P</mi></mrow><mo class="MathClass-close">]</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math> to the system <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">=</mo> <mn>0</mn></math>, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo class="MathClass-rel">|</mo><mi>L</mi><mrow><mo class="MathClass-open">(</mo><mrow><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-bin">−</mo><mi>τ</mi><mo class="MathClass-rel">|</mo> <mo class="MathClass-rel"><</mo> <mi>η</mi></math>. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>-invariant condition with the hypothesis <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi> <mo class="MathClass-rel">></mo> <mn>1</mn><mn>6</mn> <mo class="MathClass-bin">+</mo> <mn>9</mn><mi>r</mi></math> and show that the system has an integer solution. Finally, we show that the values of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> at integer zeros of C are equidistributed modulo 1 in <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>r</mi></mrow></msup></math>, requiring only that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi> <mo class="MathClass-rel">></mo> <mn>1</mn><mn>6</mn></math>. </p>
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