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The phrase "have started to improve" is correct and is commonly used in written English.
It is used to describe an ongoing process of improvement that has recently begun. Example: "Since implementing the new training program, our employees have started to improve their productivity and efficiency."
Exact(27)
However, the numbers have started to improve.
Since then, things have started to improve.
Things have started to improve, at least on the front of nuclear proliferation.
"The local committees have started to improve their political performance and their coordination.
One impact: Every other player in the nascent industry have started to improve their sites in recent days.
In recent years the police have started to improve the data they hold on antisemitism and to share anonymised details with CST.
Similar(33)
The conditions had started to improve.
And the overall economy has started to improve.
Some vigorous job-cutting had started to improve profit margins.
In recent years, the growing evidence base around endometriosis has started to improve our understanding.
But in the last few years, the situation has started to improve.
More suggestions(18)
have started to evolve
have started to implement
have launched to improve
have started to assist
have started to enforce
have started to achieve
have started to slide
have started to get
have started to study
have started to recede
have started to accept
have started to fade
have started to increase
have started to create
have started to doubt
have started to pick
have started to be
have started to leave
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com