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Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathrm {Aff}(\\mathbb {F}_{q})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>Aff<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mi>q<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for any finite field\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {F}_{q}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mi>q<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.<\/jats:p>","DOI":"10.1007\/s00454-021-00284-6","type":"journal-article","created":{"date-parts":[[2021,3,8]],"date-time":"2021-03-08T15:02:38Z","timestamp":1615215758000},"page":"1415-1428","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Number of Directions Determined by a Set in $$\\mathbb {F}_{q}^{2}$$ and Growth in $$\\mathrm {Aff}(\\mathbb {F}_{q})$$"],"prefix":"10.1007","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7966-3357","authenticated-orcid":false,"given":"Daniele","family":"Dona","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,8]]},"reference":[{"issue":"3","key":"284_CR1","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/BF02579382","volume":"6","author":"N Alon","year":"1986","unstructured":"Alon, N.: Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. 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